A splitting method for deep water with bathymetry

A splitting method for deep water with bathymetry Abstract In this article, we derive and prove the well-posedness of a deep water model that generalizes the Saut–Xu system for nonflat bottoms. Then, we present a new numerical method based on a splitting approach for studying this system. The advantage of this method is that it does not require any low-pass filter to avoid spurious oscillations. We prove a local error estimate and we show that our scheme represents a good approximation of order 1 in time. Then, we perform some numerical experiments that confirm our theoretical result, and we study three physical phenomena: the evolution of water waves over a rough bottom, the evolution of a KdV soliton when the shallowness parameter increases and the homogenization effect of rapidly varying topographies on water waves. 1. Introduction 1.1 Presentation of the problem Understanding the influence of the topography on water waves is an important issue in oceanography. Many physical phenomena are linked to the variation of the topography: shoaling, rip currents, diffraction and Bragg’s reflection. Since the direct study on the Euler equations is quite involved, several authors derived and justified asymptotic models according to different small parameters. A usual way to derive asymptotic models is to start from the Zakharov/Craig–Sulem–Sulem formulation (Zakharov, 1968; Craig et al., 1992; Craig & Sulem, 1993), which is a good formulation for irrotational water waves and to expand the Dirichlet–Neumann operator. Then, in the shallow water regime, e.g., several models were obtained such as the Saint–Venant equations or the Green–Naghdi or Boussinesq equations (see, e.g., Alvarez-Samaniego & Lannes, 2008; Iguchi, 2009; Lannes, 2013). This article addresses the influence of the bathymetry in deep water, as explained below. In this article, $$a$$ denotes the typical amplitude of the water waves, $$L$$ the typical length, $$H$$ the typical height and $$a_{\rm bott}$$ the typical amplitude of the bathymetry. Then, we introduce three parameters: $$\epsilon = \frac{a}{H}$$ the nonlinearity parameter, $$\mu = \frac{H^{2}}{L^{2}}$$ the shallowness parameter and $$\beta=\frac{a_{\rm bott}}{H}$$ the bathymetric parameter. We recall that assuming $$\mu$$ small leads to shallow water models. In deep water, which is typically the case when $$\mu$$ is of order $$1$$, it is quite common to assume that the steepness parameter $$\epsilon \sqrt{\mu} = \frac{a}{L}$$ is small. The first asymptotic model with a small steepness assumption was derived by Matsuno in two-dimensional space for a flat and nonflat bottom and weakly transverse three-dimensional water waves Matsuno, 1992, 1993. Then, Choi (1995) extended this result in three dimensions for flat bottom. Finally, Lannes & Bonneton (2009) gave a three-dimensional version in the case of a nonflat bottom. It is important to notice that these models are only formally derived. It is proved in Alvarez-Samaniego & Lannes (2008) that smooth-enough solutions to these models are close to the solutions of the water waves equations, but, to the best of our knowledge, the well-posedness of the Matsuno equations, even in the case of a flat bottom, is still an open problem. This system could be ill posed (see Ambrose et al., 2014). To avoid this difficulty, Saut & Xu (2012) developed an equivalent system to the Matsuno system, which is consistent with the water waves problem and with the same accuracy. Then, they proved that this new system is well posed. However, this model is for a flat bottom. In this article, we derive (see Section 2), use and prove the well-posedness of a generalization of the Saut–Xu system with a nonflat bottom, which is the following system: \begin{equation}\label{saut-xu-deep} \left\{ \begin{aligned} & \partial_{t} \zeta - {\mathcal{H}_{\mu}} v \epsilon \sqrt{\mu} \left( \frac{1}{2} v \partial_{x} \zeta \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) \zeta \partial_{x} v \right) = \beta \sqrt{\mu} \partial_{x} \left( B_{\mu} v \right)\\ & \partial_{t} v + \partial_{x} \zeta + \frac{3 \epsilon \sqrt{\mu}}{2} v \partial_{x} v - \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v = 0, \end{aligned} \right. \end{equation} (1.1) where (see Section 1.2 for the notations) \begin{equation*} {\mathcal{H}_{\mu}} = - \frac{\tanh(\sqrt{\mu} D)}{D} \partial_{x} \text{ and } B_{\mu} = \text{sech}( \sqrt{\mu} D) \left(b \; \text{sech}( \sqrt{\mu} D) \; \cdot \; \right)\!. \end{equation*} Many authors developed numerical approaches to study the impact of the bottom on water waves (see, e.g., Hamilton, 1977; Mei, 1985; Liu & Yue, 1998; Smith, 1998; Nachbin, 2003; Guyenne & Nicholls, 2005; Bonneton et al., 2011a,b; Cathala, 2014). However, to the best of our knowledge, when one works with deep water, there is no convergence result in the literature. After the original work of Craig & Sulem (1993) and the article of Craig et al. (2005), Guyenne & Nicholls (2007–08) developed a numerical method based on a pseudospectral method and a fourth-order Runge–Kutta scheme for the time integration. The linear terms are solved exactly, whereas the nonlinear terms are viewed as source terms. Their approach has been developed for the whole water waves equations, but we could easily adapt it to our system. However, with their scheme, we observe spurious oscillations in the wave profile that lead to instabilities. These errors seem to appear when the nonlinear part is evaluated via the Fourier transform. This is the aliasing phenomenon. Guyenne and Nicholls also observe these oscillations and, to fix it, they apply at every time step a low-pass filter. The scheme that we propose in this article avoids this low-pass filter. We present a new numerical method based on a splitting approach for studying nonlinear water waves in the presence of a bottom. We remark that the Saut–Xu system contains a dispersive part and a nonlinear transport part. Thus, the splitting method becomes an interesting alternative to solve the system since this approach is commonly used to split different physical terms (see, e.g., Ropp & Shadid, 2009). We also motivate our decomposition by the fact that, due to the pseudodifferential operator, some terms in the dispersive part may be computed efficiently using the fast Fourier transform (FFT). The transport part is computed by a Lax–Wendroff method. Various versions of the splitting method have been developed, for instance, for the nonlinear Schrodinger, the viscous Burgers’ equation and Korteweg–de Vries equations (Taha & Ablowitz, 1984; Sacchetti, 2007; Lubich, 2008; Carles, 2013; Holden et al., 2013). Thanks to this splitting, we only use a pseudospectral method for the nonlocal terms (contrary to Craig & Sulem, 1993; Guyenne & Nicholls, 2007–08), which limits the aliasing phenomenon and allows us to avoid a low-pass filter. We denote by $${\it{\Phi}}^t$$ the nonlinear flow associated with the Saut–Xu system (1.1), $${\it{\Phi}}^t_\mathcal{A}$$ and $${\it{\Phi}}^t_{\mathcal{ D}}$$, respectively, the evolution operator associated with the transport part (see equation (3.1)) and with the dispersive part (see equation (3.2)). We consider the Lie formula defined by \begin{equation} \mathcal{Y}^t = {\it{\Phi}}^t_{\mathcal{A}} {\it{\Phi}}^t_{\mathcal{D}}. \\ \end{equation} (1.2) The Saut–Xu system (1.1) is a quasilinear system. This implies derivatives losses in the proof of the convergence. In Theorem 4.6, we show that the numerical solution converges to the solution of the Saut–Xu system (1.1) in the $$H^{N+\frac{1}{2}} \times H^{N}$$-norm for initial data in $$H^{N+\frac{1}{2}-7} \times H^{N-7}({\mathbb{R}})$$, where $$N \geq 7.$$ Notice that it is not hard to generalize the present work to the Lie formula $${\it{\Phi}}^t_\mathcal{D} {\it{\Phi}}^t_{\mathcal{A}}$$. We also make the choice to prove a convergence result for a Lie splitting, but our proof can be adapted to a Strang splitting or a more complex one. Finally, notice that our scheme can be used for other equations. This article is organized as follows. In the next section, we extend the Saut–Xu system by adding a topography term and prove a local well-posedness result. We also show that the flow map $${\it{\Phi}}^t$$ is uniformly Lipschitzean. In Section 3, we split the problem and give some estimates on $${\it{\Phi}}^t_\mathcal{A}$$ and $${\it{\Phi}}^t_\mathcal{ D}$$. In Section 4, we prove a local error estimate and show that the Lie method represents a good approximation of order 1 in time (Theorem 4.6). Finally, in Section 5, we perform some numerical experiments that confirm our theoretical result, and we illustrate the results using three physical phenomena: the evolution of water waves over a rough bottom, the evolution of a KdV soliton when the shallowness parameter increases and the homogenization effect of rapidly varying topographies on water waves. 1.2 Notations and assumptions $$x$$ denotes the horizontal variable and $$z$$ the vertical variable. In this article, we only study the two-dimensional case ($$x\in {\mathbb{R}}$$). We assume that \begin{equation} \label{parameters_constraints} 0 \leq \epsilon, \beta \leq 1 \text{, } \exists \mu_{\max} > \mu_{\min} > 0 \text{, } \mu_{\max} \geq \mu \geq \mu_{\min}. \end{equation} (1.3) We denote $$\delta = \max(\epsilon, \beta)$$. We denote $${\it{\Lambda}} = (1- \partial_x^2)^{1/2} $$ and $$H^s({\mathbb{R}}) = \left\{ u \in L^2({\mathbb{R}}), ||u||_{H^2} = ||{\it{\Lambda}}^s u||_{L^2} < \infty \right\} $$, the usual Sobolev space for $$s \geq 0$$. Let $$f \in \mathcal{C}^{0} \left({\mathbb{R}} \right)$$ and $$m \in \mathbb{N}$$, such that $$ \frac{f}{1+|x|^{m}}\in L^{\infty} \left({\mathbb{R}} \right)$$. We define the Fourier multiplier $$f(D) : H^{m}\left({\mathbb{R}} \right) L^{2}\left({\mathbb{R}} \right)$$ as \begin{equation*} \forall u \in H^{m}\left({\mathbb{R}} \right) \text{, } \widehat{f(D) u}(\xi) = f(\xi) \widehat{u}(\xi). \end{equation*} $$D$$ denotes the Fourier multiplier corresponding to $$\frac{\partial_{x}}{i}$$. We denote by $$C(c_1,c_2,...)$$ a generic positive constant, strictly positive, which depends on parameters $$c_1,c_2,\cdots$$. 2. The Saut–Xu system In this part, we extend the Saut–Xu system (Saut & Xu (2012) for a nonflat bottom. Then, we give a well-posedness result that generalizes the one of Saut and Xu. The Matsuno system, which is a full dispersion model for deep waters, is an asymptotic model of the water waves equations with an accuracy of order $$\mathcal{O}\left(\delta^{2} \right)$$. Lannes & Bonneton (2009) formulated it in the following way in the presence of a nonflat topography \begin{equation}\label{matsuno} \left\{ \begin{array}{@{}l@{}} \partial_{t} \zeta - \frac{1}{\sqrt{\mu} \nu} \mathcal{H}_{\mu} v + \frac{\epsilon}{\nu} \left( {\mathcal{H}_{\mu}} \left(\zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) + \partial_{x} \left( \zeta v \right) \right) = \frac{\beta}{\nu} \partial_{x} \left(B_{\mu} v \right) \\ \partial_{t} v + \partial_{x} \zeta + \frac{\epsilon}{\nu} v \partial_{x} v -\epsilon \sqrt{\mu} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta = 0, \end{array} \right. \end{equation} (2.1) where $$\zeta = \zeta(t,x)$$ is the free surface, $$v = v(t,x)$$ is the horizontal velocity at the surface, $$\nu = \frac{\tanh(\sqrt{\mu})}{\sqrt{\mu}}$$ and $${\mathcal{H}_{\mu}}$$ and $$B_{\mu}$$ are Fourier multipliers, \begin{equation*} {\mathcal{H}_{\mu}} = - \frac{\tanh(\sqrt{\mu} D)}{D} \partial_{x} \text{ and } B_{\mu} = \text{sech}( \sqrt{\mu} D) \left(b \; \text{sech}( \sqrt{\mu} D) \; \cdot \; \right)\!, \end{equation*} and $$-1+\beta b$$ is the topography. (We erased the fluid part.) In Alvarez-Samaniego & Lannes (2008), the authors show that this model is consistent with the Zakharov/Craig–Sulem–Sulem formulation when $$\beta = 0$$, and it is not painful to generalize their result to the case when $$\beta \neq 0$$. In Saut & Xu (2012), the authors obtained a new model with the same accuracy with the Matsuno system, thanks to a nonlinear change of variables. Notice that this change of variables is inspiblack by Bona et al. (2005). The advantage of this model is that they proved a local well-posedness on large time for this new model. We follow their approach. We define new variables as follows: \begin{equation}\label{change_variable} \widetilde{v} = v + \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}} \partial_{x} \zeta \text{ and } \widetilde{\zeta} = \zeta - \frac{\epsilon \sqrt{\mu}}{4} v^{2}. \end{equation} (2.2) Then, up to terms of order $$\mathcal{O} \left( \delta^{2} \right)$$, $$\widetilde{\zeta}$$ and $$\widetilde{v}$$ satisfy (we omit the tildes for the sake of simplicity) \begin{equation}\label{saut-xu} \left\{ \begin{aligned} & \partial_{t} \zeta \left( \frac{\epsilon}{\nu} - \frac{\epsilon \sqrt{\mu}}{2} \right) v \partial_{x} \zeta - \frac{1}{\sqrt{\mu} \nu} {\mathcal{H}_{\mu}} v \frac{\epsilon}{\nu} \left( \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) \zeta \partial_{x} v \right) = \frac{\beta}{\nu} \partial_{x} \left(B_{\mu} v \right)\\ & \partial_{t} v + \left(\frac{\epsilon}{\nu} + \frac{\epsilon \sqrt{\mu}}{2} \right) v \partial_{x} v + \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta - \frac{\epsilon}{2 \nu} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v = 0. \end{aligned} \right. \end{equation} (2.3) As our motivation is the study of water waves in deep water ($$\mu$$ close to $$1$$), we assume that $$\nu=\frac{1}{\mu}$$. Hence, we study the following system, which is a variable bottom analog of the system of Saut and Xu \begin{equation*} \left\{ \begin{aligned} & \partial_{t} \zeta - {\mathcal{H}_{\mu}} v \epsilon \sqrt{\mu} \left( \frac{1}{2} v \partial_{x} \zeta \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) \zeta \partial_{x} v \right) = \beta \sqrt{\mu} \partial_{x} \left( B_{\mu} v \right)\\ & \partial_{t} v + \partial_{x} \zeta + \frac{3 \epsilon \sqrt{\mu}}{2} v \partial_{x} v - \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v = 0. \end{aligned} \right. \end{equation*} In the following, we denote $$\textbf{U} = \left(\zeta, v \right)^{t}$$ and we define the energy of the system for $$N \in \mathbb{N}$$ by \begin{equation}\label{energy} \mathcal{E}^{N} \!\! \left(\textbf{U} \right) = \frac{1}{\sqrt{\mu}} \left\lvert {\it{\Lambda}}^{N} \zeta \right\rvert_{2}^{2} + \left\lvert |D|^{\frac{1}{2}} {\it{\Lambda}}^{N} \zeta \right\rvert_{2}^{2} + \left\lvert v \right\rvert_{H^{N}}^{2}\!, \end{equation} (2.4) where $$ {\it{\Lambda}} = \sqrt{1 + |D|^2} $$ and $$D = - i \nabla $$. We also denote by $$E^{N}_{\mu}$$ the energy space related to this norm. Remark 2.1 Notice that if $$\mu$$ satisfies condition (1.3), the energy $$\mathcal{E}^{N}$$ is equivalent to the $$H^{N+\frac{1}{2}} \times H^{N}$$-norm. The main result of this section is the following local well-posedness result. Theorem 2.2 Let $$N \geq 2$$, $$\textbf{U}_{0} \in H^{N+\frac{1}{2}} \left({\mathbb{R}} \right) \times H^{N} \left({\mathbb{R}} \right)$$ and $$b \in L^{\infty} \left({\mathbb{R}} \right)$$. We assume that $$\epsilon, \beta, \mu$$ satisfy Condition (1.3) and \begin{equation*} \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} + \left\lvert b \right\rvert_{L^{\infty}} \leq M. \end{equation*} Then, there exists a time $$T_{0} = T_{0} \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right)$$ independent of $$\epsilon$$, $$\mu$$ and $$\beta$$, and a unique solution $$\textbf{U} \in \mathcal{C} \left(\left[0, \frac{T_{0}}{\delta} \right]\!, E^{N}_{\mu} \right)$$ of the system (1.1) with initial data $$\textbf{U}_{0}$$. Furthermore, we have the following energy estimate, for all $$t \in \left[0, \frac{T_{0}}{\delta} \right]$$, \begin{equation*} \mathcal{E}^{N} \!\! \left(\textbf{U}(t, \cdot) \right) \leq e^{\delta C_{0} t} \mathcal{E}^{N} \!\! \left(\textbf{U}_{0} \right)\!, \end{equation*} where $$C_{0} = C \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right)$$. Proof. We refer to Paragraph IV in Saut & Xu (2012) for a complete proof, and we focus only on the bottom contribution. For $$0 \leq \alpha \leq N$$, we denote $$\textbf{U}^{(\alpha)} = \left(\partial_{x}^{\alpha} \zeta, \partial_{x}^{\alpha} v \right)$$. Then, applying $$\partial_{x}^{\alpha}$$ to System (1.1), we get \begin{equation*} \partial_{t} \textbf{U}^{(\alpha)} + \mathcal{L} \textbf{U}^{(\alpha)} + \frac{\epsilon \sqrt{\mu}}{2} {\rm 1}\kern-0.24em{\rm I}_{\left\{ \alpha \neq 0 \right\}} \mathcal{B}[\textbf{U}] \textbf{U}^{(\alpha)} = \beta \sqrt{\mu} \left(\partial_{x} \partial_{x}^{\alpha} \left(B_{\mu} v \right)\!, 0 \right)^{t} + \epsilon \sqrt{\mu} \mathcal{G}^{\alpha}, \end{equation*} where \begin{equation*} \begin{aligned} &\mathcal{L} = \begin{pmatrix} 0 & - \frac{1}{\sqrt{\mu} \nu} {\mathcal{H}_{\mu}} \\ \partial_{x} & 0 \end{pmatrix}\!, \\ &\mathcal{B}[\textbf{U}] = \begin{pmatrix} {\mathcal{H}_{\mu}} \left( v {\mathcal{H}_{\mu}} \partial_{x} \; \cdot \; \right) + v \partial_{x} & {\mathcal{H}_{\mu}} \left( \; \cdot \; {\mathcal{H}_{\mu}} \partial_{x} \zeta \right) - \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} \\ - \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} - {\mathcal{H}_{\mu}} \partial_{x} \zeta \partial_{x} & 3 v \partial_{x} - v {\mathcal{H}_{\mu}}^{2} \partial_{x} \end{pmatrix} \end{aligned} \end{equation*} and $$\mathcal{G}^{\alpha} = (\mathcal{G}^{\alpha}_{1}, \mathcal{G}^{\alpha}_{2})^{t}$$, with \begin{equation*} \begin{aligned} &\mathcal{G}^{\alpha}_{1} = \partial_{x}^{\alpha} g(\zeta,v) - \frac{1}{2} \underset{1 \leq \gamma \leq \alpha-1}{\sum} C^{\gamma}_{\alpha} \left( {\mathcal{H}_{\mu}} ( \partial_{x}^{\gamma} v {\mathcal{H}_{\mu}} \partial_{x}^{1+\alpha-\gamma} \zeta) + \partial_{x}^{\gamma} v \partial_{x}^{1+\alpha-\gamma} \zeta \right) - \frac{1}{2} \partial_{x} \zeta ({\mathcal{H}_{\mu}}^{2}+1) \partial_{x}^{\alpha} v\\ &\mathcal{G}^{\alpha}_{2} = \frac{1}{2} \underset{1 \leq \gamma \leq \alpha-1}{\sum} C^{\gamma}_{\alpha} \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+\alpha-\gamma} \zeta + \underset{1 \leq \gamma \leq \alpha}{\sum} C^{\gamma}_{\alpha} \left( -\frac{3}{2} \partial_{x}^{\gamma} v \partial_{x}^{1+\alpha-\gamma} v + \frac{1}{2} \partial_{x}^{\gamma} v {\mathcal{H}_{\mu}}^{2} \partial_{x}^{1+\alpha-\gamma} v \right)\!, \end{aligned} \end{equation*} where \begin{equation*} g(\zeta,v) = - [{\mathcal{H}_{\mu}}, \zeta] {\mathcal{H}_{\mu}} \partial_{x} v - \zeta ({\mathcal{H}_{\mu}}^{2}+1) \partial_{x} v. \end{equation*} Then we can show, as in Saut & Xu (2012, Paragraph IV. B) (see the paragraph called Estimate on $$\mathcal{G}^{\alpha}$$), that \begin{equation}\label{estim_1} \left\lvert \mathcal{G}^{\alpha} \right\rvert_{2} + \left\lvert \left\lvert D \right\rvert^{\frac{1}{2}} \mathcal{G}^{\alpha} \right\rvert_{2} \leq C \left(\frac{1}{\mu_{\min}} \right) \mathcal{E}^{N} \! \left( \textbf{U} \right)\!. \end{equation} (2.5) Similar to Saut and Xu, we define a symmetrizer for $$\mathcal{L} + \mathcal{B}[\textbf{U}]$$ \begin{equation}\label{symmetrizer} \mathcal{S} = \begin{pmatrix} \frac{D}{\tanh(\sqrt{\mu} D)} & 0 \\ 0 & 1 \end{pmatrix}\!. \end{equation} (2.6) Notice that $$\sqrt{\left(\mathcal{S} \; \cdot, \cdot \right)}$$ is a norm equivalent to $$\sqrt{\mathcal{E}^{0}}$$. Then, as in Saut & Xu (2012, Paragraph IV. B) (see the paragraph called Estimate on II), we get \begin{equation}\label{estim_2} \left(\left( \mathcal{L} + \frac{\epsilon \sqrt{\mu}}{2} {\rm 1}\kern-0.24em{\rm I}_{\left\{ \alpha \neq 0 \right\}} \mathcal{B}[\textbf{U}] \right) \textbf{U}^{(\alpha)}, \mathcal{S} \textbf{U}^{(\alpha)} \right)\leq \epsilon C \left(\frac{1}{\mu_{\min}}, \mu_{\max} \right) \sqrt{\mathcal{E}^{2} \left(\textbf{U} \right)} \mathcal{E}^{N} \! \left(\textbf{U} \right)\!. \end{equation} (2.7) Furthermore, for the bottom contribution, we easily get \begin{equation}\label{estim_3} \left\lvert \left( \frac{D}{\tanh(\sqrt{\mu} D)} \partial_{x}^{\alpha} \zeta, \partial_{x} \partial_{x}^{\alpha} \text{sech} \left(\sqrt{\mu} D \right) \left(b \; \text{sech} \left(\sqrt{\mu} D \right) v \right) \right) \right\rvert \leq C \left(\frac{1}{\mu_{\min}} \right) \left\lvert b \right\rvert_{\infty} \mathcal{E}^{N} \! \left({\textbf{U}} \right)\!. \end{equation} (2.8) Finally, from equations (2.5), (2.7) and (2.8), we obtain \begin{equation*} \mathcal{E}^{N} \! \left(\textbf{U} \right) \leq \mathcal{E}^{N} \! \left(\textbf{U}_{0} \right) + \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max} \right) \int_{0}^{t} \left(\mathcal{E}^{N} \! \left({\textbf{U}} \right)^{\frac{3}{2}} + \mathcal{E}^{N} \! \left({\textbf{U}} \right) \right)(s) \,{\rm d}s, \end{equation*} and there exists a time $$T>0$$, such that for all $$t \in \left[0, \frac{T}{\delta} \right]$$, \begin{equation*} \mathcal{E}^{N} \! \left(\textbf{U}(t,\cdot) \right) \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, \mathcal{E}^{N} \!\! \left(\textbf{U}_{0} \right) \right)\!. \end{equation*} The energy estimate follows from the Gronwall Lemma. □ To use a Lady Windermere’s fan argument (a well-known telescopic identity used to relate global and the local error), to prove the convergence of the numerical scheme, we need a Lipschitz property for the flow of the Saut–Xu system (1.1). We first give a control of the differential of the flow with respect to the initial datum. Proposition 2.3 Let $$N \geq 2$$, $$\textbf{V}_{0} \in H^{N+\frac{1}{2}} \left({\mathbb{R}} \right) \times H^{N} \left({\mathbb{R}} \right)$$, $$\textbf{U}_{0} \in H^{N+1+\frac{1}{2}} \left({\mathbb{R}} \right) \times H^{N+1} \left({\mathbb{R}} \right)$$ and $$b \in L^{\infty} \left({\mathbb{R}} \right)$$. We assume that $$\epsilon, \beta, \mu$$ satisfy Condition (1.3) and \begin{equation*} \left\lvert \textbf{V}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} + \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+1+\frac{1}{2}} \times H^{N+1}} + \left\lvert b \right\rvert_{L^{\infty}} \leq M. \end{equation*} Then, there exists a time $$T = T \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right)$$ independent of the parameters $$\epsilon$$, $$\mu$$ and $$\beta$$ such that $$\left({\it{\Phi}}^{t} \right)' \left( \textbf{U}_{0} \right) \cdot \left(\textbf{V}_{0} \right)$$ exists on $$\left[0, \frac{T}{\delta} \right]$$. Furthermore, we have, for all $$0 \leq t \leq \frac{T}{\delta}$$, \begin{equation*} \left\lvert \left({\it{\Phi}}^{t} \right)' \left( \textbf{U}_{0} \right) \cdot \left(\textbf{V}_{0} \right) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \left\lvert \textbf{V}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}}\!. \end{equation*} Proof. We denote by $$\textbf{U}(t) = \left( \zeta(t), v(t) \right)$$ the solution of the Saut–Xu system (1.1) with initial data $$\textbf{U}_{0}$$. We denote also $$\left(\eta(t), w(t) \right) = \left({\it{\Phi}}^{t} \right)' \left( \textbf{U}_{0} \right) \cdot \left(\textbf{V}_{0} \right)$$. Then, $$\left(\eta, w \right)$$ satisfy the following system \begin{equation}\label{saut_xu_linearise} \partial_{t} \begin{pmatrix} \eta \\ w \end{pmatrix} + \mathcal{L} \begin{pmatrix} \eta \\ w \end{pmatrix} + \epsilon \sqrt{\mu} \mathcal{N}[(\zeta,v)] \partial_{x} \begin{pmatrix} \eta \\ w \end{pmatrix} + \epsilon \sqrt{\mu} \mathcal{N}[(\eta,w)] \partial_{x} \begin{pmatrix} \zeta \\ v \end{pmatrix} = \beta \sqrt{\mu} \left(\partial_{x} \left(B_{\mu} w \right), 0 \right)^{t}\!, \end{equation} (2.9) where \begin{equation*} \mathcal{L} = \begin{pmatrix} 0 & - \frac{1}{\sqrt{\mu} \nu} {\mathcal{H}_{\mu}} \\ \partial_{x} & 0 \end{pmatrix} \quad \text{ and } \quad \mathcal{N}[(\zeta,v)] = \begin{pmatrix} \frac{1}{2} {\mathcal{H}_{\mu}} \left(v {\mathcal{H}_{\mu}} \cdot \right) + \frac{1}{2} v & {\mathcal{H}_{\mu}} \left(\zeta {\mathcal{H}_{\mu}} \cdot \right) \zeta \\ - \frac{1}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} & \frac{3}{2} v - \frac{1}{2} v {\mathcal{H}_{\mu}}^{2} \end{pmatrix}\!. \end{equation*} For $$0 \leq \alpha \leq N$$, we denote $$\textbf{V}^{(\alpha)} = \left(\partial_{x}^{\alpha} \eta, \partial_{x}^{\alpha} w \right)$$. Then, applying $$\partial_{x}^{\alpha}$$ to System (2.9), we get \begin{equation*} \partial_{t} \textbf{V}^{(\alpha)} \mathcal{L} \textbf{V}^{(\alpha)} \frac{\epsilon \sqrt{\mu}}{2} {\rm 1}\kern-0.24em{\rm I}_{\left\{ \alpha \neq 0 \right\}} \left( \mathcal{B}[\textbf{U}] \textbf{V}^{(\alpha)} + \mathcal{B}[\textbf{V}] \partial_{x}^{\alpha} \textbf{U} \right) = \sqrt{\mu} \beta \begin{pmatrix} \partial_{x} \partial_{x}^{\alpha} \left(B_{\mu} w \right) \\ 0 \end{pmatrix} + \epsilon \sqrt{\mu} \mathcal{J}^{\alpha} , \end{equation*} where \begin{equation*} \begin{aligned} & \mathcal{B}[\textbf{U}] = \begin{pmatrix} {\mathcal{H}_{\mu}} \left( v {\mathcal{H}_{\mu}} \partial_{x} \; \cdot \; \right) + v \partial_{x} & {\mathcal{H}_{\mu}} \left( \; \cdot \; {\mathcal{H}_{\mu}} \partial_{x} \zeta \right) - \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} \\ - \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} - {\mathcal{H}_{\mu}} \partial_{x} \zeta \partial_{x} & 3 v \partial_{x} - v {\mathcal{H}_{\mu}}^{2} \partial_{x} \end{pmatrix}\!,\\ & \mathcal{J}^{\alpha} = - \partial_{x}^{\alpha} \left( \mathcal{N}[(\zeta,v)] \partial_{x} \begin{pmatrix} \eta \\ w \end{pmatrix} + \mathcal{N}[(\eta,w)] \partial_{x} \begin{pmatrix} \zeta \\ v \end{pmatrix} \right) + \frac{1}{2} \left( \mathcal{B}[\textbf{U}] \textbf{V}^{(\alpha)} + \mathcal{B}[\textbf{V}] \partial_{x}^{\alpha} \textbf{U} \right)\!. \end{aligned} \end{equation*} Then, we can show, as in Saut & Xu (2012, Paragraph IV. B), that \begin{equation}\label{control_J} \left\lvert \mathcal{J}^{\alpha} \right\rvert_{2} + \left\lvert \left\lvert D \right\rvert^{\frac{1}{2}} \mathcal{J}^{\alpha} \right\rvert_{2} \leq \epsilon \sqrt{\mu} C \left(\frac{1}{\mu_{\min}} \right) \mathcal{E}^{N} \! \left( {\textbf{V}} \right)\!. \end{equation} (2.10) We recall that we can symmetrize $$\mathcal{L}$$, thanks to \begin{equation*} \mathcal{S} = \begin{pmatrix} \frac{D}{\tanh(\sqrt{\mu} D)} & 0 \\ 0 & 1 \end{pmatrix}\!. \end{equation*} We define the energy associated with this symmetrizer \begin{equation*} F^{\alpha} \left( \textbf{V} \right) = \left\lvert \sqrt{\frac{D}{\tanh(\sqrt{\mu} D)}} \partial_{x}^{\alpha} \eta \right\rvert^{2}_{2} + \left\lvert \partial_{x}^{\alpha} w \right\rvert^{2}_{2} \quad \text{ and } \quad F^{N} \left( \textbf{V} \right) = \underset{0 \leq \alpha \leq N}{\sum} F^{\alpha} \left( \textbf{V} \right)\!. \end{equation*} We have, for $$\alpha \neq 0$$, \begin{equation*} \begin{aligned} \frac{\rm d}{\rm dt} F^{\alpha} \left( \textbf{V} \right) &= \epsilon \sqrt{\mu} \left( \mathcal{J}^{\alpha} , \mathcal{S} \textbf{V}^{(\alpha)} \right) - \frac{\epsilon \sqrt{\mu}}{2} \left( \left( \mathcal{B}[\textbf{U}] \textbf{V}^{(\alpha)},\mathcal{S} \textbf{V}^{(\alpha)} \right) + \left( \mathcal{B}[\textbf{V}] \partial_{x}^{\alpha} \textbf{U},\mathcal{S} \textbf{V}^{(\alpha)} \right) \right)\\ &\quad + \beta \sqrt{\mu} \left(\partial_{x} \partial_{x}^{\alpha} \left(B_{\mu} v \right),\mathcal{S} \textbf{V}^{(\alpha)} \right)\\ &= I + II + III + IIII. \end{aligned} \end{equation*} We can estimate I, thanks to estimate (2.10) and II as in Saut & Xu (2012, Paragraph IV. B). For IIII, we can proceed as in the previous theorem. For III, we get, thanks to Proposition A.1 in the Appendix, \begin{equation*} \left\lvert III \right\rvert \leq \epsilon \sqrt{\mu} \left\lvert \left( \zeta, v \right) \right\rvert_{H^{N+1+\frac{1}{2}} \times H^{N+1}} \left\lvert \left( \eta, w \right) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}}\!. \end{equation*} Then, we obtain \begin{equation*} \frac{\rm d}{\rm dt} F^{N} \left( \textbf{V} \right) \leq \delta \sqrt{\mu} C(M) \left( F^{N} \left( \textbf{V} \right) + \sqrt{F^{N} \left(\textbf{V} \right)} \right)\!, \end{equation*} and the result follows. □ Proposition 2.4 Let $$N \geq 2$$, $$\textbf{U}_{0}, \textbf{V}_{0} \in H^{N+1+\frac{1}{2}} \times H^{N+1} \left({\mathbb{R}} \right)$$ and $$b \in L^{\infty} \left({\mathbb{R}} \right)$$. We assume that $$\epsilon, \beta, \mu$$ satisfy Condition (1.3) and \begin{equation*} \left\lvert \textbf{V}_{0} \right\rvert_{H^{N+1+\frac{1}{2}} \times H^{N+1}} + \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+1+\frac{1}{2}} \times H^{N+1}} + \left\lvert b \right\rvert_{L^{\infty}} \leq M. \end{equation*} Then, there exists a time $$T$$ independent of $$\epsilon$$, $$\mu$$ and $$\beta$$ and two unique solutions $$\textbf{U}, \textbf{V}$$ of the system (1.1) on $$\left[0, \frac{T}{\delta} \right]$$ with initial data $$\textbf{U}_{0}$$ and $$\textbf{V}_{0}$$. Furthermore, we have the following Lipschitz estimate, for all $$0 \leq t \leq \frac{T}{\delta}$$, \begin{equation}\label{ConstLip} \left\lvert \textbf{U}(t, \cdot) - \textbf{V}(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq K \left\lvert \textbf{U}_{0} - \textbf{V}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}}\!, \end{equation} (2.11) where $$K = C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right)$$. Proof. The existence of $$\textbf{U}, \textbf{V}$$ and $$T$$ follow from the previous theorem. Furthermore, we have \begin{equation*} \textbf{U}(t) - \textbf{V}(t) = \int_{s=0}^{1} \left({\it{\Phi}}^{t} \right)' \left(\textbf{V}_{0} + s \left(\textbf{U}_{0} - \textbf{V}_{0} \right) \right) \cdot \left(\textbf{U}_{0} - \textbf{V}_{0} \right)\!. \end{equation*} The result follows from Proposition 2.3. □ 3. A splitting scheme In this section, we split the Saut–Xu system (1.1) and give some estimates for the subproblems. We consider, separately, the transport part \begin{equation}\label{transport_part} \left\{ \begin{array}{@{}l@{}} \partial_{t} \zeta +\frac{\epsilon \sqrt{\mu}}{2} \left( \left({\mathcal{H}_{\mu}}^{2} + 1 \right) v \right) \partial_{x} \zeta = 0\\ \partial_{t} v + \frac{3 \epsilon \sqrt{\mu}}{2} v \partial_{x} v = 0 \end{array} \right. \end{equation} (3.1) and the dispersive part \begin{equation}\label{dispersive_part} \left\{ \begin{array}{@{}l@{}} \partial_{t} \zeta - {\mathcal{H}_{\mu}} v \epsilon \sqrt{\mu} \left( \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) \zeta \partial_{x} v - \frac{1}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} v \right) = \beta \sqrt{\mu} \partial_{x} \left(B_{\mu} v \right)\\ \partial_{t} v \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v = 0. \\ \end{array} \right. \end{equation} (3.2) We denote by $${\it{\Phi}}_{\mathcal{\rm A}}^{t}$$ the flow of System (3.1) and by $${\it{\Phi}}_{\mathcal{\rm D}}^{t}$$ the flow of System (3.2). Remark 3.1 Notice that we keep the term $$\zeta \partial_{x} v$$ in the first equation and we decompose $$v \partial_{x} \zeta$$ as $$v \partial_{x} \zeta = \partial_{x} \zeta \left({\mathcal{H}_{\mu}}^{2} +1 \right)v - \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} v$$. This will be useful for the local well-posedness of the dispersive part. In the following, we prove the local existence on large time for Systems (3.1) and (3.2). 3.1 The transport equation The system (3.1) is a transport equation. Then, it is easy to get the following result. Proposition 3.2 Let $$s_{1} \geq 0$$, $$s_{2} > \frac{3}{2}$$ and $$M > 0$$. We assume that $$\epsilon, \mu$$ satisfies Condition (1.3). Then, there exists a time $$T_{1} = T_{1} \left(M,\mu_{\max} \right) >0$$, such that if \begin{equation*} \left\lvert \zeta_{0} \right\rvert_{H^{s_{1}}} + \left\lvert v_{0} \right\rvert_{H^{s_{2}}} \leq M, \end{equation*} we have a unique solution $$\left(\zeta, v \right) \in \mathcal{C} \left(\left[0, \frac{T_{1}}{\epsilon} \right], H^{s_{1}}({\mathbb{R}}) \times H^{s_{2}}({\mathbb{R}}) \right)$$, to System (3.1) with initial data $$\left(\zeta_{0}, v_{0} \right)$$. Furthermore, we have, for all $$t\leq \frac{T_{1}}{\epsilon}$$, \begin{equation}\label{estim_int1} \left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}} \leq C(M,\mu_{\max}). \end{equation} (3.3) Finally, if $$s_{2} \geq 4$$ and $$M_{1} = \underset{0 \leq t \leq \frac{T_{1}}{\epsilon}}{\max} \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}-2}}$$ then for all $$t\leq \frac{T_{1}}{\epsilon}$$, we have \begin{equation}\label{borne1} \left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}} \leq e^{\epsilon C_{1} t} |{\mathbf{U}}_0|_{H^{s_1}\times H^{s_2}}, \end{equation} (3.4) where $$C_{1}>0$$ depends on $$M_{1}$$ and $$\mu_{\max}$$. Proof. The proof follows from the fact that the quasilinear system (3.1) is symmetric. Thanks to the Coifman–Meyer estimate (see Proposition A.3) in the Appendix, we get \begin{equation*} \frac{\rm d}{\rm dt} \left(\left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}}^{2} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}}^{2} \right) \leq C \epsilon \sqrt{\mu} \left( \left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}}^{2} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}}^{2} \right)^{\frac{3}{2}}\!. \end{equation*} Then, we see that the energy is bounded uniformly with respect to $$\epsilon$$ and $$\mu$$, and we get Estimate (3.3). For the second estimate, using the same trick that in Holden et al. (2013, Lemma 3.1), we notice that, if $$s_{1} \geq 4$$, \begin{equation*} \frac{\,{\rm d}}{{\rm d}t} \left(\left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}}^{2} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}}^{2} \right) \leq \epsilon \sqrt{\mu} \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}-2}} \left( \left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}}^{2} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}}^{2} \right)\!. \end{equation*} By applying the Gronwall lemma, we get the result. □ 3.2 The dispersive equation The system (3.2) contains all the dispersive terms of the Saut–Xu system. We have the following estimate for the flow. Proposition 3.3 Let $$N \geq 2$$ and $$b \in L^{\infty}({\mathbb{R}})$$. We assume that $$\epsilon, \beta, \mu$$ satisfy Condition (1.3). Then, there exists a time $$T_{2} = T_{2} \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right)$$, such that if \begin{equation*} \left\lvert \zeta_{0} \right\rvert_{H^{N+\frac{1}{2}}} + \left\lvert v_{0} \right\rvert_{H^{N}} + \left\lvert b \right\rvert_{L^{\infty}} \leq M, \end{equation*} we have a unique solution $$\left(\zeta,v \right) \in \mathcal{C} \left(\left[0, \frac{T_{2}}{\delta} \right], H^{N+\frac{1}{2}} ({\mathbb{R}}) \times H^{N}({\mathbb{R}}) \right)$$ to the system (3.2) with initial data $$\left(\zeta_{0},v_{0} \right)$$. Furthermore, we have, for all $$t \leq \frac{T_{2}}{\delta}$$, \begin{equation}\label{estim_int2} \left\lvert \zeta(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}}} + \left\lvert v(t,\cdot) \right\rvert_{H^{N}} \leq C \left( M,\mu_{\max}, \frac{1}{\mu_{\min}} \right)\!. \end{equation} (3.5) Finally, if $$N \geq 7$$, and \begin{equation*} M_{1} = \underset{0 \leq t \leq \frac{T_{2}}{\delta}}{\max} \left( \left\lvert \zeta(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}-2}} + \left\lvert v(t,\cdot) \right\rvert_{H^{N-2}} \right)\!, \end{equation*} then for all $$t\leq \frac{T_{2}}{\delta}$$, we have \begin{equation}\label{borne2} \left\lvert \zeta(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}}} + \left\lvert v(t,\cdot) \right\rvert_{H^{N}} \leq e^{\delta C_{2} t} |{\mathbf{U}}_0|_{H^{N+1/2} \times H^N }, \end{equation} (3.6) where $$C_{2}$$ is a positive constant that depends on $$\mu_{\max}, \frac{1}{\mu_{\min}}, M_{1}$$. Proof. The proof is an adaptation of the proof of Theorem 2.2 and Saut & Xu (2012, Part IV). We notice that, in the proof of Saut and Xu, the transport part can be treated separately and does not influence the control of the other terms. Hence, we can use the same symmetrizer $$\mathcal{S}$$ that in Theorem 2.2 (see (2.6)) and we get \begin{equation*} \frac{{\rm d}}{{\rm d}t} \mathcal{E}^{N} \! \left(\zeta,v \right) \leq C \left(\frac{1}{\mu_{\min}} \right) \left(\frac{\epsilon}{\nu} \mathcal{E}^{N} \! \left(\zeta,v \right)^{\frac{3}{2}} + \frac{\beta}{\nu} \mathcal{E}^{N} \! \left(\zeta,v \right) \right)\!. \end{equation*} Then, by Remark 2.1, we get Estimate (3.5). Furthermore, we notice that we use the same trick as in Holden et al. (2013, Lemma 3.1). By keeping the same notations as in Theorem 2.2, we get from equations (2.7) and (2.8) that \begin{equation*} \frac{{\rm d}}{{\rm d}t} \mathcal{E}^{N} \! \left(\textbf{U} \right) \leq \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max} \right) \left( \left\lvert \tilde{\mathcal{G}}^{N} \right\rvert_{ H^{1/2}\times L^{2}} + \sqrt{\mathcal{E}^{2} \left(\textbf{U} \right)} + \left\lvert b \right\rvert_{\infty} \right) \mathcal{E}^{N} \! \left(\textbf{U} \right)\!, \end{equation*} where $$\tilde{\mathcal{G}}^{N} = (\tilde{\mathcal{G}}^{N}_{1}, \tilde{\mathcal{G}}^{N}_{2})^{t}$$ with \begin{equation*} \begin{aligned} &\tilde{\mathcal{G}}^{N}_{1} = - \partial_{x}^{N} \left( [{\mathcal{H}_{\mu}}, \zeta] {\mathcal{H}_{\mu}} \partial_{x} v + \zeta ({\mathcal{H}_{\mu}}^{2}+1) \partial_{x} v \right) - \frac{1}{2} \underset{1 \leq \gamma \leq N-1}{\sum} C^{\gamma}_{N} {\mathcal{H}_{\mu}} ( \partial_{x}^{\gamma} v {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta) - \frac{1}{2} \partial_{x} \zeta ({\mathcal{H}_{\mu}}^{2}+1) \partial_{x}^{N} v\\ &\tilde{\mathcal{G}}^{N}_{2} = \frac{1}{2} \underset{1 \leq \gamma \leq N-1}{\sum} C^{\gamma}_{N} \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta + \frac{1}{2} \underset{1 \leq \gamma \leq N}{\sum} C^{\gamma}_{N} \partial_{x}^{\gamma} v {\mathcal{H}_{\mu}}^{2} \partial_{x}^{1+N-\gamma} v. \end{aligned} \end{equation*} To explain how we can adapt the trick used in Holden et al. (2013, Lemma 3.1), we focus our attention to one term. For $$1 \leq \gamma \leq N-1$$, we have to control $$\left\lvert \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{L^2}$$. If $$\gamma \leq \lfloor \frac{N}{2} \rfloor$$, we get from Propositions A.4 and A.1 in the Appendix that \begin{equation*} \left\lvert \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{L^2} \leq \left\lvert \partial_{x}^{1+\gamma} \zeta \right\rvert_{H^{1}} \left\lvert {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{L^2} \leq C \left(\mu_{\max} \right) \left\lvert \zeta \right\rvert_{H^{2 + \lfloor \frac{N}{2} \rfloor}} \left\lvert \zeta \right\rvert_{H^{N}}\!, \end{equation*} whereas if $$\gamma > \lfloor \frac{N}{2} \rfloor$$, we have \begin{equation*} \left\lvert \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{L^2} \leq \left\lvert \partial_{x}^{1+\gamma} \zeta \right\rvert_{L^2} \left\lvert {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{H^1} \leq C \left(\mu_{\max} \right) \left\lvert \zeta \right\rvert_{H^{N}} \left\lvert \zeta \right\rvert_{H^{2+ \lfloor \frac{N}{2} \rfloor}}\!. \end{equation*} We can mimic this method to control the other terms of $$\tilde{\mathcal{G}}^{N}$$ and, thanks to Propositions A.1, A.2 and A.4 in the Appendix, we obtain if $$N \geq 7$$ that \begin{equation*} \frac{{\rm d}}{{\rm d}t} \mathcal{E}^{N} \! \left(\zeta,v \right) \leq \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max} \right) \left( \left\lvert \zeta(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}-2}} + \left\lvert v(t,\cdot) \right\rvert_{H^{N-2}} + \left\lvert b \right\rvert_{L^{\infty}} \right) \mathcal{E}^{N} \! \left(\zeta,v \right)\!. \end{equation*} Then, Estimate (3.6) follows. □ Remark 3.4 Under the assumption of Proposition 3.3 and if $$N \geq 7$$, we get from relations (3.4) and (3.6) that, there exists a time $$T_{3}>0$$, such that for all $$t\in \left[0, \frac{T_{3}}{\delta} \right]$$, \begin{equation*} |\mathcal{Y}^t {\mathbf{U}}_0 |_{ H^{N+1/2}\times H^{N}} \leq e^{C_{3} \delta t} |{\mathbf{U}}_0|_{H^{N+1/2}\times H^{N}}, \end{equation*} where $$C_{3} = C \left(\left\lvert {\mathbf{U}}_0 \right\rvert_{H^{N+1/2-2}\times H^{N-2}}, \mu_{\max}, \frac{1}{\mu_{\min}} \right)$$ and $$T_{3} = C \left( \left\lvert {\mathbf{U}}_0 \right\rvert_{H^{N+1/2}\times H^{N}}, \mu_{\max}, \frac{1}{\mu_{\min}} \right)$$. 4. Error estimates The goal of this part is to prove the main result of this article (Theorem 4.6). Our analysis is based on energy estimates. 4.1 The local error estimate The local error is the following quantity \begin{equation}\label{localerror} e \left(t, \textbf{U}_{0} \right) = {\it{\Phi}}^{t} \textbf{U}_{0} - \mathcal{Y}^{t} \textbf{U}_{0}. \end{equation} (4.1) Our approach is similar to the one developed in Chartier et al. (2016). We use the fact that $${\it{\Phi}}^{t} \textbf{U}_{0}$$ satisfies a symmetrizable system. Therefore, $$e$$ satisfies this system up to a remainder and then, we can control $$e$$, thanks to energy estimates. In the following, we give different technical lemmas, to control the local error. We recall that the transport operator is the operator $$\mathcal{A}$$ \begin{equation*} \mathcal{A} \left(\zeta, v \right) = - \frac{\epsilon \sqrt{\mu}}{2} \begin{pmatrix} \left(\left({\mathcal{H}_{\mu}}^{2} + 1 \right) \! v \right) \partial_{x} \zeta \\ 3 v \partial_{x} v \end{pmatrix}\!. \end{equation*} The following proposition gives an estimate of the differential of the transport operator. Lemma 4.1 Let $$s_{1}, s_{2} \geq 0$$ and $$\epsilon, \mu$$ satisfying Condition (1.3). Then, \begin{equation*} \left\lvert \mathcal{A}' (\zeta,v).(\eta,w) \right\rvert_{H^{s_{1}} \times H^{s_{2}}} \leq \epsilon C(\mu_{\max}) \left\lvert (\zeta,v) \right\rvert_{H^{s_{1}+1} \times H^{s_{2}+1}} \left\lvert (\eta,w) \right\rvert_{H^{s_{1}+1} \times H^{s_{2}+1}}\!. \end{equation*} Proof. We have \begin{equation*} \mathcal{A}' (\zeta,v).(\eta,w) = - \frac{\epsilon \sqrt{\mu}}{2} \begin{pmatrix} \left( \left({\mathcal{H}_{\mu}}^{2} + 1 \right) \! v \right) \partial_{x} \eta + \left( \left({\mathcal{H}_{\mu}}^{2} + 1 \right) \! w \right)\partial_{x} \zeta \\ 3 w \partial_{x} v+ 3 v \partial_{x} w \end{pmatrix}\!, \end{equation*} and the estimate follows from Propositions A.1 and A.4 in the Appendix. □ We can do the same for the dispersive part (using also Proposition A.1 in the Appendix). We recall that the dispersive operator is the operator $$\mathcal{D}$$ \begin{equation*} \mathcal{D}(\zeta,v) = \begin{pmatrix} {\mathcal{H}_{\mu}} v \epsilon \sqrt{\mu} \left( \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) + {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) + \zeta \partial_{x} v - \frac{1}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} v \right) - \beta \sqrt{\mu} \partial_{x} \left(B_{\mu} v \right) \\ - \partial_{x} \zeta + \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta + \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v \end{pmatrix}\!. \end{equation*} Lemma 4.2 Let $$s >0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Then, \begin{equation*} \left\lvert \mathcal{D}' (\zeta,v).(\eta,w) \right\rvert_{H^{s} \times H^{s}} \leq C(\mu_{\max}) \left(1 \beta \left\lvert b \right\rvert_{L^{\infty}} \epsilon \left\lvert (\zeta,v) \right\rvert_{H^{s+1} \times H^{s+1}} \right) \left\lvert (\eta,w) \right\rvert_{H^{s+1} \times H^{s+1}}\!. \end{equation*} Furthermore, we have to control the derivative of the flow $${\it{\Phi}}_{\mathcal{A}}^{t}$$ with respect to the initial data. We denote it by $$\left({\it{\Phi}}_{\mathcal{A}}^{t} \right)'$$. Lemma 4.3 Let $$s_{1}, s_{2} \geq 0$$, $$M > 0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Let $$\left( \zeta_{0}, v_{0} \right) \in H^{s_{1}+1} \times H^{s_{2}+1} ({\mathbb{R}^{d}})$$ such that, \begin{equation*} \left\lvert \left( \zeta_{0}, v_{0} \right) \right\rvert _{H^{s_{1}+1} \times H^{s_{2}+1}} \leq M. \end{equation*} Then, there exists a time $$T = T(M,\mu_{\max})$$, such that $$\left({\it{\Phi}}_{\mathcal{A}}^{t} \right)' \left( \zeta_{0}, v_{0} \right) \cdot \left(\eta_{0},w_{0} \right)$$ exists for all $$t \in \left[0, \frac{T}{\delta} \right]$$, and if we denote \begin{equation*} \begin{pmatrix} \eta \\ w \end{pmatrix} = \left({\it{\Phi}}_{\mathcal{A}}^{t} \right)' \left( \zeta_{0}, v_{0} \right) \cdot \left(\eta_{0},w_{0} \right) \end{equation*} for all $$0 \leq t \leq \frac{T}{\delta }$$, \begin{equation*} \left\lvert \left( \eta, w \right)(t,\cdot) \right\rvert_{H^{s_{1}} \times H^{s_{2}}} \leq \left\lvert \left( \eta_{0}, w_{0} \right) \right\rvert _{H^{s_{1}} \times H^{s_{2}}} C \left(\mu_{\max}, M \right)\!. \end{equation*} Proof. The quantity $$\left( \eta, w \right)$$ satisfies the following linear system \begin{equation*} \left\{ \begin{array}{@{}l@{}} \partial_{t} \eta + \frac{\epsilon \sqrt{\mu}}{2} \left({\mathcal{H}_{\mu}}^{2} + 1 \right) v \partial_{x} \eta + \frac{\epsilon \sqrt{\mu}}{2} \left({\mathcal{H}_{\mu}}^{2} + 1 \right) w \partial_{x} \zeta = 0,\\ \partial_{t} w + \frac{3 \epsilon \sqrt{\mu}}{2} v \partial_{x} w + \frac{3 \epsilon \sqrt{\mu}}{2} w \partial_{x} v = 0, \end{array} \right. \end{equation*} where $$\left( \zeta, v \right) = {\it{\Phi}}_{\mathcal{A}}^{t} \left( \zeta_{0}, v_{0} \right)$$. The result follows from energy estimates, the Gronwall lemma and Proposition 3.2. □ In the following, we use the fact $${\it{\Phi}}^{t}_{\mathcal{A}} \circ {\it{\Phi}}^{t}_{\mathcal{D}}$$ satisfies the Saut–Xu system (1.1) up to a remainder. The following lemma is the key point for the control of this remainder. Lemma 4.4 Let $$N \geq 2$$, $$M > 0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Let $$\textbf{U} = \left( \zeta, v \right) \in H^{N+\frac{1}{2}} \times H^{N} ({\mathbb{R}^{d}})$$ such that, \begin{equation*} \left\lvert b \right\rvert_{L^{\infty}} + \left\lvert\textbf{U} \right\rvert _{H^{N+\frac{1}{2}} \times H^{N} ({\mathbb{R}})} \leq M. \end{equation*} Then, there exists a time $$T = T \left(M, \mu_{\max}, \frac{1}{\mu_{\min}} \right) > 0$$, such that $${\it{\Phi}}^{t}_{\mathcal{A}} \left( \textbf{U} \right)$$ exists for all $$0 \leq t \leq \frac{T}{\delta}$$, and furthermore, \begin{equation*} \left\lvert \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left(\textbf{U} \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left(\textbf{U} \right) \right) \right\rvert_{H^{N-2} \times H^{N-2}} \leq \epsilon C \left(M, \mu_{\max}, \frac{1}{\mu_{\min}} \right) t. \end{equation*} Proof. The existence of $$T$$ follows from Proposition 3.2. Then, we notice that \begin{equation*} \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left(\textbf{U} \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left(\textbf{U} \right) \right) = \int_{0}^{t} \mathcal{A}' \left({\it{\Phi}}_{\mathcal{A}}^{s} \left(\textbf{U} \right) \right) \cdot \left( \left({\it{\Phi}}^{s}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left( \textbf{U} \right) \right) - \mathcal{D}' \left( {\it{\Phi}}_{\mathcal{A}}^{s} \left(\textbf{U} \right) \right) \cdot \mathcal{A} \left( {\it{\Phi}}^{s}_{\mathcal{A}} \left( \textbf{U} \right) \right)\!. \end{equation*} Using Lemmas 4.1, 4.2 and Proposition 3.2, we get \begin{align*} \left\lvert \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left(\textbf{U} \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left(\textbf{U} \right) \right) \right\rvert_{H^{N-2} \times H^{N-2}} &\leq C \left( \mu_{\max}, M \right) \int_{0}^{t} \epsilon \left\lvert \left({\it{\Phi}}^{s}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left( \textbf{U} \right) \right\rvert_{H^{N-1} \times H^{N-1}}\\ &\quad{} + \left\lvert \mathcal{A} \left( {\it{\Phi}}^{s}_{\mathcal{A}} \left( \textbf{U} \right) \right) \right\rvert_{H^{N-1} \times H^{N-1}}\!. \end{align*} Then, using Lemma 4.3, the product estimate A.4 and the expression of $$\mathcal{A}$$, we obtain \begin{equation*} \left\lvert \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left(\textbf{U} \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left(\textbf{U} \right) \right) \right\rvert_{H^{N-2} \times H^{N-2}} \leq \epsilon C \left( \mu_{\max}, M \right) \int_{0}^{t} \left\lvert \mathcal{D} \left( \textbf{U} \right) \right\rvert_{H^{N-1} \times H^{N-1}} + \left\lvert {\it{\Phi}}^{s}_{\mathcal{A}} \left( \textbf{U} \right) \right\rvert^{2}_{H^{N} \times H^{N}}\!. \end{equation*} Finally, the result follows from the expression of $$\mathcal{D}$$, the product estimate A.4 and Proposition A.1 in the Appendix. □ We can now give the main result of this part, the local error estimate. Proposition 4.5 Let $$N \geq 4$$, $$M > 0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Let $$\textbf{U}_{0} = \left( \zeta_{0}, v_{0} \right)$$, such that \begin{equation*} \left\lvert b \right\rvert_{L^{\infty}} + \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq M. \end{equation*} Then, there exists a time $$T_{4} = T_{4} \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right) > 0$$, such that the local error $$e \left(t, \textbf{U} \right)$$ defined in (4.1) exists for all $$0 \leq t \leq \frac{T_{4}}{\delta }$$, and furthermore, \begin{equation*} \left\lvert e \left(t, \textbf{U}_{0} \right) \right\rvert_{H^{N-4 + \frac{1}{2}} \times H^{N-4}} \leq \delta C_{4} t^{2}, \end{equation*} where $$C_{4} = C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right)$$. Proof. From Propositions 3.2 and 3.3, we obtain the existence of $$T$$. We denote \begin{equation*} \textbf{U}(t) = \begin{pmatrix} \zeta(t) \\ v(t) \end{pmatrix} = {\it{\Phi}}^{t} \left(\textbf{U}_{0} \right) \text{ and }\textbf{V}(t) = \begin{pmatrix} \eta(t) \\ w(t) \end{pmatrix} = {\it{\Phi}}^{t}_{\mathcal{A}} \left( {\it{\Phi}}^{t}_{\mathcal{D}} \left(\textbf{U}_{0} \right) \right)\!. \end{equation*} Then, from Theorem 2.2 and Propositions 3.2 and 3.3, we also have, for all $$0 \leq t \leq \frac{T}{\delta}$$, \begin{equation}\label{controlUV} \left\lvert \textbf{U}(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} + \left\lvert \textbf{V}(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right)\!. \end{equation} (4.2) We know that $$\left(\zeta, v \right)$$ satisfies the Saut–Xu system (1.1). Furthermore, $$\left(\eta, w \right)$$ also satisfies the Saut–Xu system (1.1) up to a remainder \begin{equation*} \partial_{t} \begin{pmatrix} \eta \\ w \end{pmatrix} = \mathcal{A} \left(\eta,w\right) + \mathcal{D} \left(\eta,w\right) + \mathcal{R}(t), \end{equation*} where $$\mathcal{R}(t) = \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left({\it{\Phi}}^{t}_{\mathcal{D}} \left(\textbf{U}_{0} \right) \right) \cdot \mathcal{D} \left({\it{\Phi}}^{t}_{\mathcal{D}} \left(\textbf{U}_{0} \right) \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left({\it{\Phi}}^{t}_{\mathcal{D}} \left(\textbf{U}_{0} \right) \right) \right)$$. Therefore, the local error $$e$$ satisfies the following system \begin{equation}\label{eq_on_e} \partial_{t} e = \begin{pmatrix} 0 & H_{\mu} \\- \partial_{x} & 0 \end{pmatrix} e + \begin{pmatrix} 0 & \beta \sqrt{\mu} B_{\mu} \\ 0 & 0 \end{pmatrix} e + \mathcal{T}_{\mu} \left(\left(\zeta,v \right), \left(\eta,w \right)\right) - \mathcal{R}(t), \end{equation} (4.3) where the operator $$\mathcal{T}_{\mu} \left(\textbf{U}, \textbf{V} \right)$$ is quadratic and satisfies the following estimate, for $$0 \leq s \leq N-1$$, \begin{equation}\label{controlnonlin} \left\lvert \mathcal{T}_{\mu} \left(\left(\zeta,v \right), \left(\eta,w \right)\right) \right\rvert_{H^{s} \times H^{s}} \leq \epsilon C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \left\lvert e \right\rvert_{H^{s+1} \times H^{s+1}}\!. \end{equation} (4.4) Then, since $$e_{|t=0}=0$$, \begin{equation*} e(t,\cdot) = \int_{0}^{t} \partial_{t} e(s,\cdot)\,{\rm d}s, \end{equation*} and since $$e$$ satisfies (4.3), we obtain, thanks to Estimates (4.2) and (4.4) and Lemma 4.4, \begin{equation}\label{firstenergyest} \left\lvert e \left(t, \cdot \right) \right\rvert_{H^{N-2} \times H^{N-2}} \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) t. \end{equation} (4.5) Furthermore, we recall that the Saut–Xu system (1.1) is symmetrizable, thanks to the symmetrizer (see Theorem 2.2) \begin{equation*} \mathcal{S} = \begin{pmatrix} \frac{D}{\tanh(\sqrt{\mu} D)} & 0 \\ 0 & 1 \end{pmatrix}\!. \end{equation*} Therefore, applying $$\mathcal{S}$$ to the system (4.3), and using the fact that $$\sqrt{\left( \mathcal{S} \cdot, \cdot \right)}$$ is a norm equivalent to the $$H^{\frac{1}{2}} \times L^{2}$$-norm, we obtain, thanks to Estimates (4.2), (4.4) and (4.5) and Lemma 4.4, \begin{equation*} \frac{\rm d}{\rm dt} \mathcal{F}(e) \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \left(\beta \mathcal{F}(e) + \epsilon t \sqrt{\mathcal{F}(e)} \right)\!, \end{equation*} where $$\mathcal{F}(e) = \sum \limits_{|\alpha| \leq N-4} \left(S \partial_{x}^{\alpha} e, \partial_{x}^{\alpha} e \right)$$. Then, we get \begin{equation*} \mathcal{F}(e)(t) \leq \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \int_{0}^{t} \mathcal{F}(e)(s) + s \sqrt{\mathcal{F}(e)(s)} \,{\rm d}s. \end{equation*} Denoting $$\mathcal{M}(t) = \underset{[0,t]}{\max} \sqrt{\mathcal{F}(e)(t)}$$, we have \begin{equation*} \mathcal{M}(t) \leq \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \int_{0}^{t} \mathcal{M}(s) + s \,{\rm d}s, \end{equation*} and the result follows from the Grönwall’s lemma. □ 4.2 Global error estimate In this part, we prove our main result. We denote by \begin{equation*} \textbf{U}_{k} = \left(\mathcal{Y}^{{\it{\Delta}} t}\right)^k \textbf{U}_{0} \end{equation*} the approximate solution and by $$ {\mathbf{U}}(t_k) := {\it{\Phi}}^{k {\it{\Delta}} t} {\mathbf{U}}_0$$ the exact solution at the time $$t_k=k {\it{\Delta}} t$$. Theorem 4.6 Let $$N \geq 7$$, $$M > 0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Let $$\textbf{U}_{0} = \left( \zeta_{0}, v_{0} \right)$$, such that \begin{equation*} \left\lvert b \right\rvert_{L^{\infty}} + \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq M. \end{equation*} Let $$\textbf{U}_{0}(t, \cdot)$$ the solution of the Saut–Xu equations (1.1) with initial data $$\textbf{U}_{0}$$ defined on $$\left[0, \frac{T}{\delta} \right]$$. Then, there exist constants $$A\gamma, \nu, {\it{\Delta}} t_0, C_0>0$$, such that for all $${\it{\Delta}} t \in ]0,{\it{\Delta}} t_0]$$ and for all $$n \in {\mathbb{N}}$$, such that $$0 \leq n {\it{\Delta}} t \leq \frac{T}{\delta }$$, \begin{equation*} | {\mathbf{U}}_n|_{H^{N + \frac{1}{2}} \times H^{N}} \leq \nu \text{ and } \left\lvert {\it{\Phi}}^{n {\it{\Delta}} t} \left( \textbf{U}_{0} \right) - \left( \mathcal{Y}^{{\it{\Delta}} t} \right)^{n} \left( \textbf{U}_{0} \right) \right\rvert_{H^{N-4 + \frac{1}{2}} \times H^{N-4}} \leq \gamma {\it{\Delta}} t. \end{equation*} Proof. The proof is based on a Lady’s Windermere’s fan argument and is similar to the one in Carles (2013) (see also Holden et al., 2013). To simplify the notations, we forget the dependence on $$\frac{1}{\mu_{\min}}$$ and $$\mu_{\max}$$ in all the constants. We denote by $$X^{N}$$ the following space \begin{equation*} X^{N} = H^{N + \frac{1}{2}} \times H^{N}. \end{equation*} Thanks to Theorem 2.2, there exists $$\rho$$ such that, for all $${t^k = k {\it{\Delta}} t } \in \left[ 0,\frac{T}{\delta } \right]$$, \begin{equation*} \left\lvert {\mathbf{U}}(t_k) \right\rvert_{X^{N}} = \left\lvert {\it{\Phi}}^{t_k} \textbf{U}_{0} \right\rvert_{X^{N}} \leq \rho. \end{equation*} We prove by induction that there exists $${\it{\Delta}} t_0, \gamma, \nu$$, such that if $$0 < {\it{\Delta}} t \leq {\it{\Delta}} t_0,$$ for all $$n \in {\mathbb{N}}$$ with $$n {\it{\Delta}} t \leq \frac{T}{\delta}$$, \begin{align*} &(i) \text{ } |{\mathbf{U}}_n-{\mathbf{U}}(t_n) |_{X^{N-4}} \leq \gamma {\it{\Delta}} t,\\ &(ii) \text{ } |{\mathbf{U}}_n|_{X^{N}} \leq e^{C_{3}(M_{1}) \delta n {\it{\Delta}} t} \left\lvert \textbf{U}_{0} \right\rvert_{X^{N}} \leq M_{0},\\ &(iii) \text{ } |{\mathbf{U}}_n|_{X^{N-2}} \leq M_{1},\\ &(iv) \text{ } |{\mathbf{U}}_n|_{X^{N-4}} \leq 2 \rho, \end{align*} with \begin{align*} &M_{1} = e^{C_{3}(2 \rho) T} M, M_{0} = e^{C_{3}(M_{1}) T} M, \gamma = T \max(K,1) C_{4} \left(e^{C_{0}(M_{0}) T} M_{0} \right)\!,\\ &{\it{\Delta}} t_{0} = \min \left(T,T_{0}(M_{0}), T_{3}(M_{0}), T_{4}(M_{0}), \frac{\rho}{\gamma} \right),K=K \left( M_{0} e^{T C_{0}(M_{0})} \right)\!, \end{align*} where $$C_{0}$$, $$T_{0}$$, $$C_{3}$$, $$T_{3}$$, $$C_{4}$$, $$T_{4}$$ and $$K$$ are constants from Theorem 2.2, Remark 3.4, Proposition 4.5 and Inequality (2.11). The above properties are satisfied for $$n=0$$. Let $$n\geq 1$$, and suppose that the induction assumptions are true for $$0 \leq k \leq n-1$$. First, we have the following telescopic series (see Carles, 2013; Holden et al., 2013) \begin{equation}\label{telecsopic_series} \textbf{U}_{n} - {\mathbf{U}}(t_n) = \underset{0 \leq k \leq n-1}{\sum} {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} \mathcal{Y}^{{\it{\Delta}} t} \textbf{U}_{k} - {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} {\it{\Phi}}^{{\it{\Delta}} t} \textbf{U}_{k}. \end{equation} (4.6) For $$k\leq n-2$$, since $$\mathcal{Y}^{{\it{\Delta}} t} {\mathbf{U}}_k = {\mathbf{U}}_{k+1}$$, using the induction assumption (ii), we have \begin{equation*} |\mathcal{Y}^{{\it{\Delta}} t} {\mathbf{U}}_k|_{X^{N-3}} \leq M_{0}, \end{equation*} and from Theorem 2.2, we get \begin{equation*} |{\it{\Phi}}^{{\it{\Delta}} t} {\mathbf{U}}_k|_{X^{N-3}} \leq e^{C_{0}(M_{0}) T} M_{0}. \end{equation*} Therefore, from Proposition 2.4 and up to replacing $$K=K \left( M_{0}\, e^{T C_{0}(M_{0})} \right)$$ with $$\max(K,1)$$, we obtain, for $$k \leq n-1$$ and $$n {\it{\Delta}} t \leq \frac{T}{\delta}$$, \begin{equation*} \left\lvert {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} \mathcal{Y}^{{\it{\Delta}} t} \textbf{U}_{k} - {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} {\it{\Phi}}^{{\it{\Delta}} t} \textbf{U}_{k} \right\rvert_{X^{N-4}} \leq K \left\lvert \mathcal{Y}^{{\it{\Delta}} t} {\mathbf{U}}_k- {\it{\Phi}}^{{\it{\Delta}} t} {\mathbf{U}}(t_k) \right\rvert_{X^{N-4}}\!. \end{equation*} Then, using Proposition 4.5 and Inequality (ii), we infer \begin{equation*} \left\lvert {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} \mathcal{Y}^{{\it{\Delta}} t} \textbf{U}_{k} - {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} {\it{\Phi}}^{{\it{\Delta}} t} \textbf{U}_{k} \right\rvert_{X^{N-4}} \leq \delta C_{4} \left( e^{C_{0}(M_{0}) T} M_{0} \right) K ({\it{\Delta}} t)^2. \end{equation*} Therefore, using the telescopic series (4.6), we get \begin{equation*} |\textbf{U}_{n} - {\mathbf{U}}(t_n)|_{X^{N-4}} \leq n C_{4} \left( e^{C_{0}(M_{0}) T} M_{0} \right) K \delta ({\it{\Delta}} t)^2 \leq C_{4} \left( e^{C_{0}(M_{0}) T} M_{0} \right) K T {\it{\Delta}} t. \end{equation*} For Estimate (ii), using Remark 3.4 and the induction assumptions (iii) and (ii), we have \begin{equation*} \left\lvert \textbf{U}_{n} \right\rvert_{X^{N}} = \left\lvert \mathcal{Y}^{{\it{\Delta}} t} \left( \textbf{U}_{n-1} \right) \right\rvert_{X^{N}} \leq e^{\delta C_{3}(M_{1}) {\it{\Delta}} t} \left\lvert \textbf{U}_{n-1} \right\rvert_{X^{N}} \leq e^{C_{3}(M_{1}) \delta n {\it{\Delta}} t} \left\lvert \textbf{U}_{0} \right\rvert_{X^{N}} \leq M_{0}. \end{equation*} We get Estimate (iii) in the same way, using the induction assumptions (iv) and (iii). Finally, for Estimate (iv), using (i), we have \begin{align*} \left\lvert \textbf{U}_{n} \right\rvert_{X^{N-4}} \leq \left\lvert \textbf{U}_{n} - \textbf{U}(t_{n}) \right\rvert_{X^{N-4}} + \left\lvert \textbf{U}(t_{n}) \right\rvert_{X^{N-4}} \leq \gamma {\it{\Delta}} t \rho \leq 2 \rho. \end{align*} □ 5. Numerical experiments The aim of this section is to numerically verify the Lie method convergence rate in $$\mathcal{O}({\it{\Delta}} t)$$ for the Saut–Xu system (1.1) and to illustrate some physical phenomena. In other works and particularly on the whole water waves problem (see, e.g., Craig & Sulem, 1993; Nicholls & Reitich, 2001; Guyenne & Nicholls, 2007–08; and references therein), several authors use a discrete Fourier transform even for the transport part. They observe spurious oscillations in the wave profile that lead to instabilities. These errors seem to appear when they evaluate the nonlinear part via Fourier transform, because additional terms appear in the approximation, which is the aliasing phenomenon. To fix this problem, they apply at every time step a low-pass filter. The main interest of our scheme is that we do not need one, because we use a finite difference method to approximate the nonlinear part. For the dispersive equation (3.2), we use the forward Euler discretization in time and for the spatial discretization, we consider the FFT implemented in Matlab. In this scheme, the interval $$[0,1]$$ is discretized by $$N$$ equidistant points, with spacing $${\it{\Delta}} x = 1/N$$. The spatial grid points are then given by $$x_j = j/N$$, $$j=0,...,N$$. Therefore, if $$u_j(t)$$ denotes the approximate solution to $$u(t,x_j)$$, the discrete Fourier transform of the sequence $$ \left\lbrace u_j \right\rbrace_{j = 0}^{N-1} $$ is defined by \begin{equation*} \hat{u}(k) = \mathcal{F}^d_k (u_j) = \sum_{j=0}^{N-1} u_j\, e^{-2i \pi j k /N }, \end{equation*} for $$k = 0,\cdots, N-1$$, and the inverse discrete Fourier transform is given by \begin{equation*} u_j = \mathcal{F}_{j}^{-d}( \hat{u}_k ) = \frac{1}{N} \sum_{k = 0}^{N -1} \hat{u}_k \,e^{ 2 i \pi k x_j}, \end{equation*} for $$j = 0, \cdots,N-1 $$. Here, $$\mathcal{F}^d$$ denotes the discrete Fourier transform and $$\mathcal{F}^{-d}$$ its inverse. Then, in what follows, the numerical scheme to solve (3.2) is given by \begin{equation} \begin{pmatrix} \zeta^{n+1}_j \\ v_j^{n+1} \end{pmatrix} = \begin{pmatrix} \zeta^{n}_j \\ v_j^{n} \end{pmatrix} - {\it{\Delta}} t \begin{pmatrix} F_j^{n} + S_j^n \\ G_j^{n} \end{pmatrix}\!, \label{subeq1} \end{equation} (5.1) where $$S_j^n = \beta \sqrt{\mu} \mathcal{F}^{-d}_j(i k \mathcal{F}^d_k (B_\mu v^n_j ))$$ and $$F_j^n = I_1 + I_2 $$ with \begin{align*} I_1 &= \mathcal{F}^{-d}_j \left( i \, \tanh\left(\sqrt{\mu} k\right) \left( -1 + \frac{\epsilon \sqrt{\mu}}{2} \mathcal{F}^d_k\left( v^n_j \mathcal{F}^{-d}_j\left(k \tanh\left(\sqrt{\mu k} \right) \hat{\zeta}^n_k \right) \right) \right.\right.\\ &\left.\left.+ \epsilon \sqrt{\mu} \mathcal{F}^d_k\left( \zeta^n_j \mathcal{F}^{-d}_j\left(k \tanh(\sqrt{\mu k} ) \hat{v}^n_k \right) \right) \right) \right) \end{align*} \begin{equation*} I_2 = \zeta_j^n \mathcal{F}^{-d}_j\left( i k \hat{v}^n_k \right) + \frac{1}{2} \mathcal{F}^{-d} \left( i k \hat{\zeta}^n_k \right) \mathcal{F}_j^{-d} \left( \tanh(\sqrt{\mu} k )^2 \hat{v}^n_k \right)\!. \end{equation*} To approximate equation (3.1), we use the following finite difference scheme \begin{equation} \begin{pmatrix} \zeta^{n+1}_j \\ v_j^{n+1} \end{pmatrix} = \begin{pmatrix} \zeta^{n}_j \\ v_j^{n} \end{pmatrix} - {\it{\Delta}} t \frac{\epsilon \sqrt{\mu} }{2} \begin{pmatrix} G^n_1 \\ 3 G^n_2 \end{pmatrix}\!, \label{subeq2} \end{equation} (5.2) where \begin{equation*} G_1 = w_j^n \frac{\zeta^n_{j+1}-\zeta^n_{j-1}}{2 {\it{\Delta}} x} - \frac{{\it{\Delta}} t}{2 {\it{\Delta}} x^2} (w^n_j)^2 \left( \zeta^n_{j-1}-2\zeta^n_j+\zeta^n_{j+1} \right) \end{equation*} with $$w_j^n = - \mathcal{F}^{-d}_j \left( \tanh( \sqrt{\mu} k)^2 \hat{v}^n_k \right) + v_j^n$$ and $$ G_2^n = \frac{(v^n_{j+1})^2 - (v^n_{j-1})^2 }{2 {\it{\Delta}} x} - \frac{{\it{\Delta}} t}{2 {\it{\Delta}} x^2} \left( v^n_{j+1/2} \left( (v^n_{j+1})^2 - (v^n_{j})^2 \right) - v^n_{j-1/2} \left( (v^n_j)^2 - (v^n_{j-1} )^2 \right) \right) $$ with $$v^n_{j \pm 1/2} = \frac{ v^n_j + v^n_{j \pm 1} }{2}. $$ We remarked that for our numerical simulations, it is not necessary to decompose the term $$v \partial_x \zeta$$ (see Remark 3.1) to get the numerical convergence. Indeed, it seems that since the time step is chosen very small, we obtain a solution of the dispersive equation for each iteration. In this case, we do not need to evaluate the term $$\partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} v$$. To ensure the validity of our numerical simulations, we have to be careful of the numerical instability, as to why the time and the space steps are chosen in a way that the following condition is satisfied: \begin{equation} \label{CFL} |v | \frac{{\it{\Delta}} t}{{\it{\Delta}} x} < 1. \end{equation} (5.3) 5.1 Example $$1$$: Convergence curve In this example, we consider the following initial data: \begin{equation*} \zeta_0(x) = {\mathrm{sech}} \left(\frac{\sqrt{3}}{2} x \right), \quad v_0= \zeta_0, \end{equation*} with two different bathymetries: a bump ($$b(x) = \cos(x)$$) and a ripple bottom $$ b(x) = \left\{ \begin{array}{@{}l@{}} 0.5 - \frac{1}{18} (x-8)^2 \mbox{ if } 5 \leq x \leq 11 \\[6pt] 0 \mbox{ otherwise}. \end{array} \right. $$ Note that to avoid numerical reflections due to the boundaries and justify the use of the FFT, we decide to take rapidly decreasing initial data. Figures 1 and 2 display the evolution for different times of the free surface $$\zeta$$ for these two test cases. We decided to take $$\epsilon = 0.1, \mu = 1, \beta = \frac{1}{2}$$, $$N = 2^8, {\it{\Delta}} x = 2L/N, T = 10$$, where N is the mesh modes number, $$L=30$$ the length of the domain and $$T$$ the final time. Note that the time step $${\it{\Delta}} t $$ is chosen iteratively in a way that the CFL condition (5.3) is satisfied. Fig. 1. View largeDownload slide Upper: Evolution of the free surface for different times t = 2.5, 5, 7.5 and 10. Lower: Bottom topography and initial condition. Fig. 1. View largeDownload slide Upper: Evolution of the free surface for different times t = 2.5, 5, 7.5 and 10. Lower: Bottom topography and initial condition. Fig. 2. View largeDownload slide Upper: Evolution of the free surface for different times t = 2.5, 5, 7,5 and 10. Lower: Bottom topography and initial condition. Fig. 2. View largeDownload slide Upper: Evolution of the free surface for different times t = 2.5, 5, 7,5 and 10. Lower: Bottom topography and initial condition. Figures 3 displays the convergence curve for this example. We plot the logarithm of the error (in norm $$H^1 \times L^2$$) in function of the logarithm of the time step $${\it{\Delta}} t.$$ The convergence numerical order is then given by the slope of this curve. For reference, a small line (the dashed line) of slope 1 is added in this figure. We see that the numerical rate of convergence is greater than 1. Fig. 3. View largeDownload slide Convergence curve (for the $$H^1 \times L^2$$-norm) for the Lie method for two bottoms: bump (lower line) and ripple bottom (higher line) for $$T=10.$$ Fig. 3. View largeDownload slide Convergence curve (for the $$H^1 \times L^2$$-norm) for the Lie method for two bottoms: bump (lower line) and ripple bottom (higher line) for $$T=10.$$ 5.2 Example $$2$$: Nonsmooth topographies In this example, we study the evolution of water waves over a rough bottom. This problem is still a mathematical issue. Many models derived from the Euler equations suppose that the bathymetry is smooth. Even worse, a nonsmooth bathymetry introduces singular terms in these models. This issue is particularly easy to see for shallow water models. To handle this, Hamilton (1977) and Nachbin (2003) used a conformal mapping to derive long wave models. Notice also the work of Cathala (2014) who derived alternatives Saint-Venant equations and Boussinesq systems with nonsmooth topographies that do not involve any singular terms. We notice that the Saut–Xu equations (1.1) can handle a nonsmooth topography (see Theorem (2.2)) and our numerical scheme too (see Theorem (4.6)). In the following, we give an example with a nonsmooth bathymetry. We consider the following initial conditions and bathymetry \begin{equation*} \zeta_{0}(x) = v_{0}(x) = e^{-x^{2}} \text{ and } b(x)=\frac{\beta}{4}(1+\tanh(100(x-2)))(1-\tanh(100(x-8))). \end{equation*} We decided to take $$\mu =1$$, $$\beta=0.5$$, $$\epsilon=0.1$$, $$N = 2^8, {\it{\Delta}} x = 2L/N$$, where N is the mesh modes number and $$L=20$$ the length of the domain. The time step $${\it{\Delta}} t $$ is chosen iteratively in a way that the CFL condition (5.3) is satisfied. Figure 4 displays the evolution of the surface $$\zeta$$ for different times over the bottom. Fig. 4. View largeDownload slide Evolution of the free surface (higher lines) for different times t = 0, 3, 6, 9 and 12 over a rough bottom (lower line). Fig. 4. View largeDownload slide Evolution of the free surface (higher lines) for different times t = 0, 3, 6, 9 and 12 over a rough bottom (lower line). 5.3 Example $$3$$: Boussinesq regime In Section 3, we crucially use the fact that $$\mu$$ is bounded from below. In this example, we test our scheme for small values of $$\mu$$ (also called the shallow water regime). We show that our scheme is still valid even if we do not have a proof of the convergence of our scheme in this regime. In the shallow water regime, there is a huge literature for asymptotic models (see, e.g., Lannes, 2013). Among all these asymptotic models, we have the KdV equation. It is a model obtained under the Boussinesq regime, i.e., when $$\epsilon = \mu$$, $$\beta = 0$$ and $$\mu$$ small. In the following, we formally derive a KdV equation from the Saut–Xu equations, and we give numerical simulations in this setting. We recall that without the assumption $$\nu = \frac{1}{\mu}$$, the Saut–Xu equations are given by the system (2.3). Notice also that \begin{equation}\label{dt_Hmu} \mathcal{H}_{\mu} = - \sqrt{\mu} \partial_{x} - \frac{1}{3} \mu^{\frac{3}{2}} \partial_{x}^{3} + \mathcal{O}(\mu^{2}). \end{equation} (5.4) Then, if we assume that $$\mu = \epsilon$$, $$\nu=1$$ (since $$\nu \sim 1$$ if $$\mu$$ is small) and we drop all the terms of order $$\mathcal{O}(\mu^{2})$$ in System (2.3), we obtain the following equations \begin{equation}\label{saut-xu_boussi} \left\{ \begin{aligned} &\partial_{t} \zeta + \partial_{x} v + \mu v \partial_{x} \zeta - \frac{1}{2} \mu^{\frac{3}{2}} v \partial_{x} \zeta + \frac{1}{3} \mu \partial_{x}^{3} v + \mu \zeta \partial_{x} v =0,\\ &\partial_{t} v + \partial_{x} \zeta + \mu v \partial_{x} v + \mu^{\frac{3}{2}} \frac{1}{2} v \partial_{x} v= 0. \end{aligned} \right. \end{equation} (5.5) Formally, the solutions of this system are close to the solutions of (2.3) with an accuracy of order $$\mathcal{O}(\mu^{2})$$. Notice that this system is not a standard Boussinesq system (in the sense of Bona et al. (2002) or Lannes (2013)) because of our nonlinear change of variables (2.2). Using the approach developed in Schneider & Wayne (2012), Bona et al. (2005) and Alvarez-Samaniego & Lannes (2008) (see also Lannes, 2013, Part 7.1.1), we can check that, formally, the following KdV equation is an asymptotic model of the system (5.5) \begin{equation}\label{kdv_eq} \partial_{t} f + \frac{3}{2} f \partial_{x} f + \frac{1}{6} \partial_{x}^{3} f = 0. \end{equation} (5.6) This means that if we solve (5.5) with the initial data $$\left(f_{0}, f_{0} \right)$$ and (5.6) with the initial datum $$f_{0}$$, the solution $$\left(\zeta, v \right)(t,x)$$ of (5.5) is close to $$\left(f,f \right) (\mu t,x-t)$$. Furthermore, if we take $$f_{0}(x) = \alpha \text{sech}^{2} \left(\sqrt{\frac{3}{4} \alpha} x \right)$$, the solution $$f$$ of the KdV equation with this initial datum is the soliton $$f(t,x) = f_{0}(x-ct)$$ with $$c=\frac{\alpha}{2}$$. Hence, in this case, the solutions of (5.5) and (2.3) are close to a soliton. In the following, we check that the solution to (1.1) is indeed close to the KdV solution when $$\mu$$ is small. We simulate one soliton. We took $$v_0(x) = \zeta_0(x) = {\mathrm{sech}}^2 \left(\frac{\sqrt{3}}{2} x \right)$$, $$\epsilon = \mu = 0.01$$, $$\alpha = 1$$ and the final time is $$T=10$$. We decided to take $$N = 2^9, {\it{\Delta}} x = 2L/N$$, where N is the mesh modes number and $$L=30$$ the length of the domain. The time step $${\it{\Delta}} t $$ is chosen iteratively in a way that the CFL condition (5.3) is satisfied. Figure 5 represents the evolution of this soliton at different times. Hence, our scheme is still valid when $$\mu$$ is small. Fig. 5. View largeDownload slide Evolution of the soliton at different times $$t=0, 3, 6, 9$$ ($$\epsilon = 0.01$$). Fig. 5. View largeDownload slide Evolution of the soliton at different times $$t=0, 3, 6, 9$$ ($$\epsilon = 0.01$$). In deep water ($$\mu$$ not small), the KdV approximation ceases to be a good approximation. To get some insight on the range of validity of the KdV approximation, we compare in Fig. 6 the solution of (1.1) with the exact soliton after a time $$T=10$$ for various values of $$\mu$$. We took the same numerical parameters that were taken before. We notice that even for $$\epsilon = \mu=0.1$$ and a final time $$T=\frac{1}{\mu}$$, the KdV approximation remains a good approximation of the Saut–Xu system. Fig. 6. View largeDownload slide Difference after a time $$T=10$$ between a real soliton and a soliton generated by our scheme with the same initial data and for different values of $$\epsilon=\mu$$. Abscissa: value of $$\epsilon$$; Ordinate: quotient of the difference after a final time $$T=10$$ by the maximum of the soliton. Fig. 6. View largeDownload slide Difference after a time $$T=10$$ between a real soliton and a soliton generated by our scheme with the same initial data and for different values of $$\epsilon=\mu$$. Abscissa: value of $$\epsilon$$; Ordinate: quotient of the difference after a final time $$T=10$$ by the maximum of the soliton. 5.4 Example $$4$$: Rapidly varying topographies In this example, we study the evolution of water waves over a rapidly varying periodic bottom. We assume that $$\mu =1$$. This problem is linked to the Bragg reflection phenomenon (see, e.g., Mei, 1985; Liu & Yue, 1998; Guyenne & Nicholls, 2007–08). We take \begin{equation}\label{ini_data_rapidly_bot} \zeta_{0} = v_{0} = \text{sech}^{2}\left(\frac{\sqrt{3}}{2} x \right) \text{ and } b(x)=\cos(\alpha x). \end{equation} (5.7) We decided to take $$N = 2^9, {\it{\Delta}} x = 2L/N$$, where N is the mesh mode number and $$L=30$$ the length of the domain. The time step $${\it{\Delta}} t $$ is chosen iteratively in a way that the CFL condition (5.3) is satisfied. Figure 7 displays the difference between the case of a flat bottom and the case of a bottom of the form $$b(x)=\cos(\alpha x)$$ for different values of $$\alpha$$. Figure 8 compares the evolution of water waves when we take $$\alpha=10$$ (blue line) and when we take $$b(x)=0$$ (red line). We observe a homogenization effect when $$\alpha$$ is large. It seems that when $$\alpha$$ goes to infinity, a solution of the Saut–Xu equations converges to a solution of the Saut–Xu equations with a flat bottom (corresponding to the mean of $$b$$). Notice that this result is different from what we could see in the literature (e.g., Chupin, 2012; Craig et al., 2012), since we take a bottom of the form $$b(x)=\cos (\alpha x)$$ and not of the form $$b(x)= \frac{1}{\alpha} \cos(\alpha x)$$. Our numerical simulations suggest therefore a homogenization effect for large amplitude bottom variations that has not been investigated so far. Fig. 7. View largeDownload slide Difference between a water wave over a rapidly varying topography $$b(x)=\cos(\alpha x)$$ and a water wave over a flat bottom. Abscissa: value of $$\alpha$$; Ordinate: quotient of the difference after a final time $$T=10$$ by the maximum of $$\zeta_{0}$$. Fig. 7. View largeDownload slide Difference between a water wave over a rapidly varying topography $$b(x)=\cos(\alpha x)$$ and a water wave over a flat bottom. Abscissa: value of $$\alpha$$; Ordinate: quotient of the difference after a final time $$T=10$$ by the maximum of $$\zeta_{0}$$. Fig. 8. View largeDownload slide Comparison between the evolution of a water wave over a bottom of the form $$b(x)=\cos(10x)$$ (lower line) and the evolution of a water wave over a flat bottom after a time T = 10. $$\epsilon =0.05$$, $$\beta=0.5$$. The two different surfaces are very close. Fig. 8. View largeDownload slide Comparison between the evolution of a water wave over a bottom of the form $$b(x)=\cos(10x)$$ (lower line) and the evolution of a water wave over a flat bottom after a time T = 10. $$\epsilon =0.05$$, $$\beta=0.5$$. The two different surfaces are very close. Funding ANR project Dyficolti [ANR-13-BS01-0003 to B.M.]. References Alinhac, S. & Gérard, P. ( 1991 ) Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs Actuels. [Current Scholarship]. InterEditions. Paris, France : Éditions du Centre National de la Recherche Scientifique (CNRS), Meudon . Alvarez-Samaniego, B. & Lannes, D. ( 2008 ) Large time existence for 3D water-waves and asymptotics. Invent. 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Google Scholar CrossRef Search ADS Appendix In this part, we give some estimate for the operator $${\mathcal{H}_{\mu}}$$ and some standard product and commutator estimates. For the estimates for $${\mathcal{H}_{\mu}}$$, we refer to Saut & Xu (2012, Part III). For the other estimates, we refer to Alinhac & Gérard (1991) and Lannes (2006). We recall that $${\mathcal{H}_{\mu}}$$ is defined by \begin{equation*} {\mathcal{H}_{\mu}} = - \frac{\tanh(\sqrt{\mu} D)}{D} \partial_{x}. \end{equation*} First, we show that $${\mathcal{H}_{\mu}}$$ is a zero-order operator. Proposition A.1 Let $$s \geq 0$$ and $$\mu$$ satisfying Condition (1.3). Then, \begin{equation*} \left\lvert {\mathcal{H}_{\mu}} u \right\rvert_{H^{s}} \leq C \left(\mu_{\max} \right) \left\lvert u \right\rvert_{H^{s}}\!. \end{equation*} Furthermore, for all $$s \geq r \geq 0$$, \begin{equation*} \left\lvert \left( {\mathcal{H}_{\mu}}^{2} + 1 \right) u \right\rvert_{H^{s}} \leq C_{r} \left(\frac{1}{\mu_{\min}} \right) \left\lvert u \right\rvert_{H^{r}}\!. \end{equation*} Then, we give a commutator estimate for $${\mathcal{H}_{\mu}}$$. Proposition A.2 Let $$s \geq 0$$, $$t_{0} > \frac{1}{2}$$, $$r \geq 0$$ and $$\mu$$ satisfying Condition (1.3). Then, \begin{equation*} \left\lvert \left[{\mathcal{H}_{\mu}}, a \right] u \right\rvert_{2} \leq C \left\lvert a \right\rvert_{H^{t_{0}}} \left\lvert f \right\rvert_{2}, \end{equation*} \begin{equation*} \left\lvert |\xi|^{s} \widehat{\left[{\mathcal{H}_{\mu}}, a \right] u} \right\rvert_{2} \leq C \left(\frac{1}{\mu_{\min}} \right) \left\lvert a \right\rvert_{H^{r+s}} \left\lvert \frac{\left(1+|\xi| \right)^{t_{0}}}{|\xi|^{r}} \widehat{u} \right\rvert_{2} \end{equation*} and \begin{equation*} \left\lvert |\xi|^{s} \widehat{\left[{\mathcal{H}_{\mu}}, a \right] u} \right\rvert_{2} \leq C \left(\frac{1}{\mu_{\min}} \right) \left\lvert a \right\rvert_{H^{r+s+t_{0}}} \left\lvert \frac{1}{|\xi|^{r}} \widehat{u} \right\rvert_{2}. \end{equation*} We recall the well-known Coifman–Meyer estimate. We recall also that $${\it{\Lambda}}$$ is the Fourier multiplier $${\it{\Lambda}} = \sqrt{1+D^{2}}$$. Proposition A.3 Let $$s > \frac{3}{2}$$, $$u \in H^{s}({\mathbb{R}})$$ and $$v \in H^{s-1}({\mathbb{R}})$$. Then, we have the following commutator estimate \begin{equation*} \left\lvert \left[ {\it{\Lambda}}^{s}, u \right] v \right\rvert_{2} \leq C \left\lvert u \right\rvert_{H^{s}} \left\lvert v \right\rvert_{H^{s-1}}\!. \end{equation*} We recall also the following product estimate. Proposition A.4 Let $$s_{1}, s_{2},s$$ such that $$s_{1} + s_{2} \geq 0$$, $$s \leq \min \left( s_{1}, s_{2} \right)$$ and $$s < s_{1} + s_{2} - \frac{1}{2}$$. Let $$u \in H^{s_{1}}({\mathbb{R}})$$ and $$v \in H^{s_{2}}({\mathbb{R}})$$. Then, \begin{equation*} \left\lvert u v \right\rvert_{H^{s}} \leq C \left\lvert u \right\rvert_{H^{s_{1}}} \left\lvert v \right\rvert_{H^{s_{2}}}\!. \end{equation*} © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. 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A splitting method for deep water with bathymetry

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Oxford University Press
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© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0272-4979
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1464-3642
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10.1093/imanum/drx034
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Abstract

Abstract In this article, we derive and prove the well-posedness of a deep water model that generalizes the Saut–Xu system for nonflat bottoms. Then, we present a new numerical method based on a splitting approach for studying this system. The advantage of this method is that it does not require any low-pass filter to avoid spurious oscillations. We prove a local error estimate and we show that our scheme represents a good approximation of order 1 in time. Then, we perform some numerical experiments that confirm our theoretical result, and we study three physical phenomena: the evolution of water waves over a rough bottom, the evolution of a KdV soliton when the shallowness parameter increases and the homogenization effect of rapidly varying topographies on water waves. 1. Introduction 1.1 Presentation of the problem Understanding the influence of the topography on water waves is an important issue in oceanography. Many physical phenomena are linked to the variation of the topography: shoaling, rip currents, diffraction and Bragg’s reflection. Since the direct study on the Euler equations is quite involved, several authors derived and justified asymptotic models according to different small parameters. A usual way to derive asymptotic models is to start from the Zakharov/Craig–Sulem–Sulem formulation (Zakharov, 1968; Craig et al., 1992; Craig & Sulem, 1993), which is a good formulation for irrotational water waves and to expand the Dirichlet–Neumann operator. Then, in the shallow water regime, e.g., several models were obtained such as the Saint–Venant equations or the Green–Naghdi or Boussinesq equations (see, e.g., Alvarez-Samaniego & Lannes, 2008; Iguchi, 2009; Lannes, 2013). This article addresses the influence of the bathymetry in deep water, as explained below. In this article, $$a$$ denotes the typical amplitude of the water waves, $$L$$ the typical length, $$H$$ the typical height and $$a_{\rm bott}$$ the typical amplitude of the bathymetry. Then, we introduce three parameters: $$\epsilon = \frac{a}{H}$$ the nonlinearity parameter, $$\mu = \frac{H^{2}}{L^{2}}$$ the shallowness parameter and $$\beta=\frac{a_{\rm bott}}{H}$$ the bathymetric parameter. We recall that assuming $$\mu$$ small leads to shallow water models. In deep water, which is typically the case when $$\mu$$ is of order $$1$$, it is quite common to assume that the steepness parameter $$\epsilon \sqrt{\mu} = \frac{a}{L}$$ is small. The first asymptotic model with a small steepness assumption was derived by Matsuno in two-dimensional space for a flat and nonflat bottom and weakly transverse three-dimensional water waves Matsuno, 1992, 1993. Then, Choi (1995) extended this result in three dimensions for flat bottom. Finally, Lannes & Bonneton (2009) gave a three-dimensional version in the case of a nonflat bottom. It is important to notice that these models are only formally derived. It is proved in Alvarez-Samaniego & Lannes (2008) that smooth-enough solutions to these models are close to the solutions of the water waves equations, but, to the best of our knowledge, the well-posedness of the Matsuno equations, even in the case of a flat bottom, is still an open problem. This system could be ill posed (see Ambrose et al., 2014). To avoid this difficulty, Saut & Xu (2012) developed an equivalent system to the Matsuno system, which is consistent with the water waves problem and with the same accuracy. Then, they proved that this new system is well posed. However, this model is for a flat bottom. In this article, we derive (see Section 2), use and prove the well-posedness of a generalization of the Saut–Xu system with a nonflat bottom, which is the following system: \begin{equation}\label{saut-xu-deep} \left\{ \begin{aligned} & \partial_{t} \zeta - {\mathcal{H}_{\mu}} v \epsilon \sqrt{\mu} \left( \frac{1}{2} v \partial_{x} \zeta \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) \zeta \partial_{x} v \right) = \beta \sqrt{\mu} \partial_{x} \left( B_{\mu} v \right)\\ & \partial_{t} v + \partial_{x} \zeta + \frac{3 \epsilon \sqrt{\mu}}{2} v \partial_{x} v - \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v = 0, \end{aligned} \right. \end{equation} (1.1) where (see Section 1.2 for the notations) \begin{equation*} {\mathcal{H}_{\mu}} = - \frac{\tanh(\sqrt{\mu} D)}{D} \partial_{x} \text{ and } B_{\mu} = \text{sech}( \sqrt{\mu} D) \left(b \; \text{sech}( \sqrt{\mu} D) \; \cdot \; \right)\!. \end{equation*} Many authors developed numerical approaches to study the impact of the bottom on water waves (see, e.g., Hamilton, 1977; Mei, 1985; Liu & Yue, 1998; Smith, 1998; Nachbin, 2003; Guyenne & Nicholls, 2005; Bonneton et al., 2011a,b; Cathala, 2014). However, to the best of our knowledge, when one works with deep water, there is no convergence result in the literature. After the original work of Craig & Sulem (1993) and the article of Craig et al. (2005), Guyenne & Nicholls (2007–08) developed a numerical method based on a pseudospectral method and a fourth-order Runge–Kutta scheme for the time integration. The linear terms are solved exactly, whereas the nonlinear terms are viewed as source terms. Their approach has been developed for the whole water waves equations, but we could easily adapt it to our system. However, with their scheme, we observe spurious oscillations in the wave profile that lead to instabilities. These errors seem to appear when the nonlinear part is evaluated via the Fourier transform. This is the aliasing phenomenon. Guyenne and Nicholls also observe these oscillations and, to fix it, they apply at every time step a low-pass filter. The scheme that we propose in this article avoids this low-pass filter. We present a new numerical method based on a splitting approach for studying nonlinear water waves in the presence of a bottom. We remark that the Saut–Xu system contains a dispersive part and a nonlinear transport part. Thus, the splitting method becomes an interesting alternative to solve the system since this approach is commonly used to split different physical terms (see, e.g., Ropp & Shadid, 2009). We also motivate our decomposition by the fact that, due to the pseudodifferential operator, some terms in the dispersive part may be computed efficiently using the fast Fourier transform (FFT). The transport part is computed by a Lax–Wendroff method. Various versions of the splitting method have been developed, for instance, for the nonlinear Schrodinger, the viscous Burgers’ equation and Korteweg–de Vries equations (Taha & Ablowitz, 1984; Sacchetti, 2007; Lubich, 2008; Carles, 2013; Holden et al., 2013). Thanks to this splitting, we only use a pseudospectral method for the nonlocal terms (contrary to Craig & Sulem, 1993; Guyenne & Nicholls, 2007–08), which limits the aliasing phenomenon and allows us to avoid a low-pass filter. We denote by $${\it{\Phi}}^t$$ the nonlinear flow associated with the Saut–Xu system (1.1), $${\it{\Phi}}^t_\mathcal{A}$$ and $${\it{\Phi}}^t_{\mathcal{ D}}$$, respectively, the evolution operator associated with the transport part (see equation (3.1)) and with the dispersive part (see equation (3.2)). We consider the Lie formula defined by \begin{equation} \mathcal{Y}^t = {\it{\Phi}}^t_{\mathcal{A}} {\it{\Phi}}^t_{\mathcal{D}}. \\ \end{equation} (1.2) The Saut–Xu system (1.1) is a quasilinear system. This implies derivatives losses in the proof of the convergence. In Theorem 4.6, we show that the numerical solution converges to the solution of the Saut–Xu system (1.1) in the $$H^{N+\frac{1}{2}} \times H^{N}$$-norm for initial data in $$H^{N+\frac{1}{2}-7} \times H^{N-7}({\mathbb{R}})$$, where $$N \geq 7.$$ Notice that it is not hard to generalize the present work to the Lie formula $${\it{\Phi}}^t_\mathcal{D} {\it{\Phi}}^t_{\mathcal{A}}$$. We also make the choice to prove a convergence result for a Lie splitting, but our proof can be adapted to a Strang splitting or a more complex one. Finally, notice that our scheme can be used for other equations. This article is organized as follows. In the next section, we extend the Saut–Xu system by adding a topography term and prove a local well-posedness result. We also show that the flow map $${\it{\Phi}}^t$$ is uniformly Lipschitzean. In Section 3, we split the problem and give some estimates on $${\it{\Phi}}^t_\mathcal{A}$$ and $${\it{\Phi}}^t_\mathcal{ D}$$. In Section 4, we prove a local error estimate and show that the Lie method represents a good approximation of order 1 in time (Theorem 4.6). Finally, in Section 5, we perform some numerical experiments that confirm our theoretical result, and we illustrate the results using three physical phenomena: the evolution of water waves over a rough bottom, the evolution of a KdV soliton when the shallowness parameter increases and the homogenization effect of rapidly varying topographies on water waves. 1.2 Notations and assumptions $$x$$ denotes the horizontal variable and $$z$$ the vertical variable. In this article, we only study the two-dimensional case ($$x\in {\mathbb{R}}$$). We assume that \begin{equation} \label{parameters_constraints} 0 \leq \epsilon, \beta \leq 1 \text{, } \exists \mu_{\max} > \mu_{\min} > 0 \text{, } \mu_{\max} \geq \mu \geq \mu_{\min}. \end{equation} (1.3) We denote $$\delta = \max(\epsilon, \beta)$$. We denote $${\it{\Lambda}} = (1- \partial_x^2)^{1/2} $$ and $$H^s({\mathbb{R}}) = \left\{ u \in L^2({\mathbb{R}}), ||u||_{H^2} = ||{\it{\Lambda}}^s u||_{L^2} < \infty \right\} $$, the usual Sobolev space for $$s \geq 0$$. Let $$f \in \mathcal{C}^{0} \left({\mathbb{R}} \right)$$ and $$m \in \mathbb{N}$$, such that $$ \frac{f}{1+|x|^{m}}\in L^{\infty} \left({\mathbb{R}} \right)$$. We define the Fourier multiplier $$f(D) : H^{m}\left({\mathbb{R}} \right) L^{2}\left({\mathbb{R}} \right)$$ as \begin{equation*} \forall u \in H^{m}\left({\mathbb{R}} \right) \text{, } \widehat{f(D) u}(\xi) = f(\xi) \widehat{u}(\xi). \end{equation*} $$D$$ denotes the Fourier multiplier corresponding to $$\frac{\partial_{x}}{i}$$. We denote by $$C(c_1,c_2,...)$$ a generic positive constant, strictly positive, which depends on parameters $$c_1,c_2,\cdots$$. 2. The Saut–Xu system In this part, we extend the Saut–Xu system (Saut & Xu (2012) for a nonflat bottom. Then, we give a well-posedness result that generalizes the one of Saut and Xu. The Matsuno system, which is a full dispersion model for deep waters, is an asymptotic model of the water waves equations with an accuracy of order $$\mathcal{O}\left(\delta^{2} \right)$$. Lannes & Bonneton (2009) formulated it in the following way in the presence of a nonflat topography \begin{equation}\label{matsuno} \left\{ \begin{array}{@{}l@{}} \partial_{t} \zeta - \frac{1}{\sqrt{\mu} \nu} \mathcal{H}_{\mu} v + \frac{\epsilon}{\nu} \left( {\mathcal{H}_{\mu}} \left(\zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) + \partial_{x} \left( \zeta v \right) \right) = \frac{\beta}{\nu} \partial_{x} \left(B_{\mu} v \right) \\ \partial_{t} v + \partial_{x} \zeta + \frac{\epsilon}{\nu} v \partial_{x} v -\epsilon \sqrt{\mu} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta = 0, \end{array} \right. \end{equation} (2.1) where $$\zeta = \zeta(t,x)$$ is the free surface, $$v = v(t,x)$$ is the horizontal velocity at the surface, $$\nu = \frac{\tanh(\sqrt{\mu})}{\sqrt{\mu}}$$ and $${\mathcal{H}_{\mu}}$$ and $$B_{\mu}$$ are Fourier multipliers, \begin{equation*} {\mathcal{H}_{\mu}} = - \frac{\tanh(\sqrt{\mu} D)}{D} \partial_{x} \text{ and } B_{\mu} = \text{sech}( \sqrt{\mu} D) \left(b \; \text{sech}( \sqrt{\mu} D) \; \cdot \; \right)\!, \end{equation*} and $$-1+\beta b$$ is the topography. (We erased the fluid part.) In Alvarez-Samaniego & Lannes (2008), the authors show that this model is consistent with the Zakharov/Craig–Sulem–Sulem formulation when $$\beta = 0$$, and it is not painful to generalize their result to the case when $$\beta \neq 0$$. In Saut & Xu (2012), the authors obtained a new model with the same accuracy with the Matsuno system, thanks to a nonlinear change of variables. Notice that this change of variables is inspiblack by Bona et al. (2005). The advantage of this model is that they proved a local well-posedness on large time for this new model. We follow their approach. We define new variables as follows: \begin{equation}\label{change_variable} \widetilde{v} = v + \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}} \partial_{x} \zeta \text{ and } \widetilde{\zeta} = \zeta - \frac{\epsilon \sqrt{\mu}}{4} v^{2}. \end{equation} (2.2) Then, up to terms of order $$\mathcal{O} \left( \delta^{2} \right)$$, $$\widetilde{\zeta}$$ and $$\widetilde{v}$$ satisfy (we omit the tildes for the sake of simplicity) \begin{equation}\label{saut-xu} \left\{ \begin{aligned} & \partial_{t} \zeta \left( \frac{\epsilon}{\nu} - \frac{\epsilon \sqrt{\mu}}{2} \right) v \partial_{x} \zeta - \frac{1}{\sqrt{\mu} \nu} {\mathcal{H}_{\mu}} v \frac{\epsilon}{\nu} \left( \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) \zeta \partial_{x} v \right) = \frac{\beta}{\nu} \partial_{x} \left(B_{\mu} v \right)\\ & \partial_{t} v + \left(\frac{\epsilon}{\nu} + \frac{\epsilon \sqrt{\mu}}{2} \right) v \partial_{x} v + \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta - \frac{\epsilon}{2 \nu} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v = 0. \end{aligned} \right. \end{equation} (2.3) As our motivation is the study of water waves in deep water ($$\mu$$ close to $$1$$), we assume that $$\nu=\frac{1}{\mu}$$. Hence, we study the following system, which is a variable bottom analog of the system of Saut and Xu \begin{equation*} \left\{ \begin{aligned} & \partial_{t} \zeta - {\mathcal{H}_{\mu}} v \epsilon \sqrt{\mu} \left( \frac{1}{2} v \partial_{x} \zeta \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) \zeta \partial_{x} v \right) = \beta \sqrt{\mu} \partial_{x} \left( B_{\mu} v \right)\\ & \partial_{t} v + \partial_{x} \zeta + \frac{3 \epsilon \sqrt{\mu}}{2} v \partial_{x} v - \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v = 0. \end{aligned} \right. \end{equation*} In the following, we denote $$\textbf{U} = \left(\zeta, v \right)^{t}$$ and we define the energy of the system for $$N \in \mathbb{N}$$ by \begin{equation}\label{energy} \mathcal{E}^{N} \!\! \left(\textbf{U} \right) = \frac{1}{\sqrt{\mu}} \left\lvert {\it{\Lambda}}^{N} \zeta \right\rvert_{2}^{2} + \left\lvert |D|^{\frac{1}{2}} {\it{\Lambda}}^{N} \zeta \right\rvert_{2}^{2} + \left\lvert v \right\rvert_{H^{N}}^{2}\!, \end{equation} (2.4) where $$ {\it{\Lambda}} = \sqrt{1 + |D|^2} $$ and $$D = - i \nabla $$. We also denote by $$E^{N}_{\mu}$$ the energy space related to this norm. Remark 2.1 Notice that if $$\mu$$ satisfies condition (1.3), the energy $$\mathcal{E}^{N}$$ is equivalent to the $$H^{N+\frac{1}{2}} \times H^{N}$$-norm. The main result of this section is the following local well-posedness result. Theorem 2.2 Let $$N \geq 2$$, $$\textbf{U}_{0} \in H^{N+\frac{1}{2}} \left({\mathbb{R}} \right) \times H^{N} \left({\mathbb{R}} \right)$$ and $$b \in L^{\infty} \left({\mathbb{R}} \right)$$. We assume that $$\epsilon, \beta, \mu$$ satisfy Condition (1.3) and \begin{equation*} \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} + \left\lvert b \right\rvert_{L^{\infty}} \leq M. \end{equation*} Then, there exists a time $$T_{0} = T_{0} \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right)$$ independent of $$\epsilon$$, $$\mu$$ and $$\beta$$, and a unique solution $$\textbf{U} \in \mathcal{C} \left(\left[0, \frac{T_{0}}{\delta} \right]\!, E^{N}_{\mu} \right)$$ of the system (1.1) with initial data $$\textbf{U}_{0}$$. Furthermore, we have the following energy estimate, for all $$t \in \left[0, \frac{T_{0}}{\delta} \right]$$, \begin{equation*} \mathcal{E}^{N} \!\! \left(\textbf{U}(t, \cdot) \right) \leq e^{\delta C_{0} t} \mathcal{E}^{N} \!\! \left(\textbf{U}_{0} \right)\!, \end{equation*} where $$C_{0} = C \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right)$$. Proof. We refer to Paragraph IV in Saut & Xu (2012) for a complete proof, and we focus only on the bottom contribution. For $$0 \leq \alpha \leq N$$, we denote $$\textbf{U}^{(\alpha)} = \left(\partial_{x}^{\alpha} \zeta, \partial_{x}^{\alpha} v \right)$$. Then, applying $$\partial_{x}^{\alpha}$$ to System (1.1), we get \begin{equation*} \partial_{t} \textbf{U}^{(\alpha)} + \mathcal{L} \textbf{U}^{(\alpha)} + \frac{\epsilon \sqrt{\mu}}{2} {\rm 1}\kern-0.24em{\rm I}_{\left\{ \alpha \neq 0 \right\}} \mathcal{B}[\textbf{U}] \textbf{U}^{(\alpha)} = \beta \sqrt{\mu} \left(\partial_{x} \partial_{x}^{\alpha} \left(B_{\mu} v \right)\!, 0 \right)^{t} + \epsilon \sqrt{\mu} \mathcal{G}^{\alpha}, \end{equation*} where \begin{equation*} \begin{aligned} &\mathcal{L} = \begin{pmatrix} 0 & - \frac{1}{\sqrt{\mu} \nu} {\mathcal{H}_{\mu}} \\ \partial_{x} & 0 \end{pmatrix}\!, \\ &\mathcal{B}[\textbf{U}] = \begin{pmatrix} {\mathcal{H}_{\mu}} \left( v {\mathcal{H}_{\mu}} \partial_{x} \; \cdot \; \right) + v \partial_{x} & {\mathcal{H}_{\mu}} \left( \; \cdot \; {\mathcal{H}_{\mu}} \partial_{x} \zeta \right) - \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} \\ - \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} - {\mathcal{H}_{\mu}} \partial_{x} \zeta \partial_{x} & 3 v \partial_{x} - v {\mathcal{H}_{\mu}}^{2} \partial_{x} \end{pmatrix} \end{aligned} \end{equation*} and $$\mathcal{G}^{\alpha} = (\mathcal{G}^{\alpha}_{1}, \mathcal{G}^{\alpha}_{2})^{t}$$, with \begin{equation*} \begin{aligned} &\mathcal{G}^{\alpha}_{1} = \partial_{x}^{\alpha} g(\zeta,v) - \frac{1}{2} \underset{1 \leq \gamma \leq \alpha-1}{\sum} C^{\gamma}_{\alpha} \left( {\mathcal{H}_{\mu}} ( \partial_{x}^{\gamma} v {\mathcal{H}_{\mu}} \partial_{x}^{1+\alpha-\gamma} \zeta) + \partial_{x}^{\gamma} v \partial_{x}^{1+\alpha-\gamma} \zeta \right) - \frac{1}{2} \partial_{x} \zeta ({\mathcal{H}_{\mu}}^{2}+1) \partial_{x}^{\alpha} v\\ &\mathcal{G}^{\alpha}_{2} = \frac{1}{2} \underset{1 \leq \gamma \leq \alpha-1}{\sum} C^{\gamma}_{\alpha} \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+\alpha-\gamma} \zeta + \underset{1 \leq \gamma \leq \alpha}{\sum} C^{\gamma}_{\alpha} \left( -\frac{3}{2} \partial_{x}^{\gamma} v \partial_{x}^{1+\alpha-\gamma} v + \frac{1}{2} \partial_{x}^{\gamma} v {\mathcal{H}_{\mu}}^{2} \partial_{x}^{1+\alpha-\gamma} v \right)\!, \end{aligned} \end{equation*} where \begin{equation*} g(\zeta,v) = - [{\mathcal{H}_{\mu}}, \zeta] {\mathcal{H}_{\mu}} \partial_{x} v - \zeta ({\mathcal{H}_{\mu}}^{2}+1) \partial_{x} v. \end{equation*} Then we can show, as in Saut & Xu (2012, Paragraph IV. B) (see the paragraph called Estimate on $$\mathcal{G}^{\alpha}$$), that \begin{equation}\label{estim_1} \left\lvert \mathcal{G}^{\alpha} \right\rvert_{2} + \left\lvert \left\lvert D \right\rvert^{\frac{1}{2}} \mathcal{G}^{\alpha} \right\rvert_{2} \leq C \left(\frac{1}{\mu_{\min}} \right) \mathcal{E}^{N} \! \left( \textbf{U} \right)\!. \end{equation} (2.5) Similar to Saut and Xu, we define a symmetrizer for $$\mathcal{L} + \mathcal{B}[\textbf{U}]$$ \begin{equation}\label{symmetrizer} \mathcal{S} = \begin{pmatrix} \frac{D}{\tanh(\sqrt{\mu} D)} & 0 \\ 0 & 1 \end{pmatrix}\!. \end{equation} (2.6) Notice that $$\sqrt{\left(\mathcal{S} \; \cdot, \cdot \right)}$$ is a norm equivalent to $$\sqrt{\mathcal{E}^{0}}$$. Then, as in Saut & Xu (2012, Paragraph IV. B) (see the paragraph called Estimate on II), we get \begin{equation}\label{estim_2} \left(\left( \mathcal{L} + \frac{\epsilon \sqrt{\mu}}{2} {\rm 1}\kern-0.24em{\rm I}_{\left\{ \alpha \neq 0 \right\}} \mathcal{B}[\textbf{U}] \right) \textbf{U}^{(\alpha)}, \mathcal{S} \textbf{U}^{(\alpha)} \right)\leq \epsilon C \left(\frac{1}{\mu_{\min}}, \mu_{\max} \right) \sqrt{\mathcal{E}^{2} \left(\textbf{U} \right)} \mathcal{E}^{N} \! \left(\textbf{U} \right)\!. \end{equation} (2.7) Furthermore, for the bottom contribution, we easily get \begin{equation}\label{estim_3} \left\lvert \left( \frac{D}{\tanh(\sqrt{\mu} D)} \partial_{x}^{\alpha} \zeta, \partial_{x} \partial_{x}^{\alpha} \text{sech} \left(\sqrt{\mu} D \right) \left(b \; \text{sech} \left(\sqrt{\mu} D \right) v \right) \right) \right\rvert \leq C \left(\frac{1}{\mu_{\min}} \right) \left\lvert b \right\rvert_{\infty} \mathcal{E}^{N} \! \left({\textbf{U}} \right)\!. \end{equation} (2.8) Finally, from equations (2.5), (2.7) and (2.8), we obtain \begin{equation*} \mathcal{E}^{N} \! \left(\textbf{U} \right) \leq \mathcal{E}^{N} \! \left(\textbf{U}_{0} \right) + \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max} \right) \int_{0}^{t} \left(\mathcal{E}^{N} \! \left({\textbf{U}} \right)^{\frac{3}{2}} + \mathcal{E}^{N} \! \left({\textbf{U}} \right) \right)(s) \,{\rm d}s, \end{equation*} and there exists a time $$T>0$$, such that for all $$t \in \left[0, \frac{T}{\delta} \right]$$, \begin{equation*} \mathcal{E}^{N} \! \left(\textbf{U}(t,\cdot) \right) \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, \mathcal{E}^{N} \!\! \left(\textbf{U}_{0} \right) \right)\!. \end{equation*} The energy estimate follows from the Gronwall Lemma. □ To use a Lady Windermere’s fan argument (a well-known telescopic identity used to relate global and the local error), to prove the convergence of the numerical scheme, we need a Lipschitz property for the flow of the Saut–Xu system (1.1). We first give a control of the differential of the flow with respect to the initial datum. Proposition 2.3 Let $$N \geq 2$$, $$\textbf{V}_{0} \in H^{N+\frac{1}{2}} \left({\mathbb{R}} \right) \times H^{N} \left({\mathbb{R}} \right)$$, $$\textbf{U}_{0} \in H^{N+1+\frac{1}{2}} \left({\mathbb{R}} \right) \times H^{N+1} \left({\mathbb{R}} \right)$$ and $$b \in L^{\infty} \left({\mathbb{R}} \right)$$. We assume that $$\epsilon, \beta, \mu$$ satisfy Condition (1.3) and \begin{equation*} \left\lvert \textbf{V}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} + \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+1+\frac{1}{2}} \times H^{N+1}} + \left\lvert b \right\rvert_{L^{\infty}} \leq M. \end{equation*} Then, there exists a time $$T = T \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right)$$ independent of the parameters $$\epsilon$$, $$\mu$$ and $$\beta$$ such that $$\left({\it{\Phi}}^{t} \right)' \left( \textbf{U}_{0} \right) \cdot \left(\textbf{V}_{0} \right)$$ exists on $$\left[0, \frac{T}{\delta} \right]$$. Furthermore, we have, for all $$0 \leq t \leq \frac{T}{\delta}$$, \begin{equation*} \left\lvert \left({\it{\Phi}}^{t} \right)' \left( \textbf{U}_{0} \right) \cdot \left(\textbf{V}_{0} \right) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \left\lvert \textbf{V}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}}\!. \end{equation*} Proof. We denote by $$\textbf{U}(t) = \left( \zeta(t), v(t) \right)$$ the solution of the Saut–Xu system (1.1) with initial data $$\textbf{U}_{0}$$. We denote also $$\left(\eta(t), w(t) \right) = \left({\it{\Phi}}^{t} \right)' \left( \textbf{U}_{0} \right) \cdot \left(\textbf{V}_{0} \right)$$. Then, $$\left(\eta, w \right)$$ satisfy the following system \begin{equation}\label{saut_xu_linearise} \partial_{t} \begin{pmatrix} \eta \\ w \end{pmatrix} + \mathcal{L} \begin{pmatrix} \eta \\ w \end{pmatrix} + \epsilon \sqrt{\mu} \mathcal{N}[(\zeta,v)] \partial_{x} \begin{pmatrix} \eta \\ w \end{pmatrix} + \epsilon \sqrt{\mu} \mathcal{N}[(\eta,w)] \partial_{x} \begin{pmatrix} \zeta \\ v \end{pmatrix} = \beta \sqrt{\mu} \left(\partial_{x} \left(B_{\mu} w \right), 0 \right)^{t}\!, \end{equation} (2.9) where \begin{equation*} \mathcal{L} = \begin{pmatrix} 0 & - \frac{1}{\sqrt{\mu} \nu} {\mathcal{H}_{\mu}} \\ \partial_{x} & 0 \end{pmatrix} \quad \text{ and } \quad \mathcal{N}[(\zeta,v)] = \begin{pmatrix} \frac{1}{2} {\mathcal{H}_{\mu}} \left(v {\mathcal{H}_{\mu}} \cdot \right) + \frac{1}{2} v & {\mathcal{H}_{\mu}} \left(\zeta {\mathcal{H}_{\mu}} \cdot \right) \zeta \\ - \frac{1}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} & \frac{3}{2} v - \frac{1}{2} v {\mathcal{H}_{\mu}}^{2} \end{pmatrix}\!. \end{equation*} For $$0 \leq \alpha \leq N$$, we denote $$\textbf{V}^{(\alpha)} = \left(\partial_{x}^{\alpha} \eta, \partial_{x}^{\alpha} w \right)$$. Then, applying $$\partial_{x}^{\alpha}$$ to System (2.9), we get \begin{equation*} \partial_{t} \textbf{V}^{(\alpha)} \mathcal{L} \textbf{V}^{(\alpha)} \frac{\epsilon \sqrt{\mu}}{2} {\rm 1}\kern-0.24em{\rm I}_{\left\{ \alpha \neq 0 \right\}} \left( \mathcal{B}[\textbf{U}] \textbf{V}^{(\alpha)} + \mathcal{B}[\textbf{V}] \partial_{x}^{\alpha} \textbf{U} \right) = \sqrt{\mu} \beta \begin{pmatrix} \partial_{x} \partial_{x}^{\alpha} \left(B_{\mu} w \right) \\ 0 \end{pmatrix} + \epsilon \sqrt{\mu} \mathcal{J}^{\alpha} , \end{equation*} where \begin{equation*} \begin{aligned} & \mathcal{B}[\textbf{U}] = \begin{pmatrix} {\mathcal{H}_{\mu}} \left( v {\mathcal{H}_{\mu}} \partial_{x} \; \cdot \; \right) + v \partial_{x} & {\mathcal{H}_{\mu}} \left( \; \cdot \; {\mathcal{H}_{\mu}} \partial_{x} \zeta \right) - \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} \\ - \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} - {\mathcal{H}_{\mu}} \partial_{x} \zeta \partial_{x} & 3 v \partial_{x} - v {\mathcal{H}_{\mu}}^{2} \partial_{x} \end{pmatrix}\!,\\ & \mathcal{J}^{\alpha} = - \partial_{x}^{\alpha} \left( \mathcal{N}[(\zeta,v)] \partial_{x} \begin{pmatrix} \eta \\ w \end{pmatrix} + \mathcal{N}[(\eta,w)] \partial_{x} \begin{pmatrix} \zeta \\ v \end{pmatrix} \right) + \frac{1}{2} \left( \mathcal{B}[\textbf{U}] \textbf{V}^{(\alpha)} + \mathcal{B}[\textbf{V}] \partial_{x}^{\alpha} \textbf{U} \right)\!. \end{aligned} \end{equation*} Then, we can show, as in Saut & Xu (2012, Paragraph IV. B), that \begin{equation}\label{control_J} \left\lvert \mathcal{J}^{\alpha} \right\rvert_{2} + \left\lvert \left\lvert D \right\rvert^{\frac{1}{2}} \mathcal{J}^{\alpha} \right\rvert_{2} \leq \epsilon \sqrt{\mu} C \left(\frac{1}{\mu_{\min}} \right) \mathcal{E}^{N} \! \left( {\textbf{V}} \right)\!. \end{equation} (2.10) We recall that we can symmetrize $$\mathcal{L}$$, thanks to \begin{equation*} \mathcal{S} = \begin{pmatrix} \frac{D}{\tanh(\sqrt{\mu} D)} & 0 \\ 0 & 1 \end{pmatrix}\!. \end{equation*} We define the energy associated with this symmetrizer \begin{equation*} F^{\alpha} \left( \textbf{V} \right) = \left\lvert \sqrt{\frac{D}{\tanh(\sqrt{\mu} D)}} \partial_{x}^{\alpha} \eta \right\rvert^{2}_{2} + \left\lvert \partial_{x}^{\alpha} w \right\rvert^{2}_{2} \quad \text{ and } \quad F^{N} \left( \textbf{V} \right) = \underset{0 \leq \alpha \leq N}{\sum} F^{\alpha} \left( \textbf{V} \right)\!. \end{equation*} We have, for $$\alpha \neq 0$$, \begin{equation*} \begin{aligned} \frac{\rm d}{\rm dt} F^{\alpha} \left( \textbf{V} \right) &= \epsilon \sqrt{\mu} \left( \mathcal{J}^{\alpha} , \mathcal{S} \textbf{V}^{(\alpha)} \right) - \frac{\epsilon \sqrt{\mu}}{2} \left( \left( \mathcal{B}[\textbf{U}] \textbf{V}^{(\alpha)},\mathcal{S} \textbf{V}^{(\alpha)} \right) + \left( \mathcal{B}[\textbf{V}] \partial_{x}^{\alpha} \textbf{U},\mathcal{S} \textbf{V}^{(\alpha)} \right) \right)\\ &\quad + \beta \sqrt{\mu} \left(\partial_{x} \partial_{x}^{\alpha} \left(B_{\mu} v \right),\mathcal{S} \textbf{V}^{(\alpha)} \right)\\ &= I + II + III + IIII. \end{aligned} \end{equation*} We can estimate I, thanks to estimate (2.10) and II as in Saut & Xu (2012, Paragraph IV. B). For IIII, we can proceed as in the previous theorem. For III, we get, thanks to Proposition A.1 in the Appendix, \begin{equation*} \left\lvert III \right\rvert \leq \epsilon \sqrt{\mu} \left\lvert \left( \zeta, v \right) \right\rvert_{H^{N+1+\frac{1}{2}} \times H^{N+1}} \left\lvert \left( \eta, w \right) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}}\!. \end{equation*} Then, we obtain \begin{equation*} \frac{\rm d}{\rm dt} F^{N} \left( \textbf{V} \right) \leq \delta \sqrt{\mu} C(M) \left( F^{N} \left( \textbf{V} \right) + \sqrt{F^{N} \left(\textbf{V} \right)} \right)\!, \end{equation*} and the result follows. □ Proposition 2.4 Let $$N \geq 2$$, $$\textbf{U}_{0}, \textbf{V}_{0} \in H^{N+1+\frac{1}{2}} \times H^{N+1} \left({\mathbb{R}} \right)$$ and $$b \in L^{\infty} \left({\mathbb{R}} \right)$$. We assume that $$\epsilon, \beta, \mu$$ satisfy Condition (1.3) and \begin{equation*} \left\lvert \textbf{V}_{0} \right\rvert_{H^{N+1+\frac{1}{2}} \times H^{N+1}} + \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+1+\frac{1}{2}} \times H^{N+1}} + \left\lvert b \right\rvert_{L^{\infty}} \leq M. \end{equation*} Then, there exists a time $$T$$ independent of $$\epsilon$$, $$\mu$$ and $$\beta$$ and two unique solutions $$\textbf{U}, \textbf{V}$$ of the system (1.1) on $$\left[0, \frac{T}{\delta} \right]$$ with initial data $$\textbf{U}_{0}$$ and $$\textbf{V}_{0}$$. Furthermore, we have the following Lipschitz estimate, for all $$0 \leq t \leq \frac{T}{\delta}$$, \begin{equation}\label{ConstLip} \left\lvert \textbf{U}(t, \cdot) - \textbf{V}(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq K \left\lvert \textbf{U}_{0} - \textbf{V}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}}\!, \end{equation} (2.11) where $$K = C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right)$$. Proof. The existence of $$\textbf{U}, \textbf{V}$$ and $$T$$ follow from the previous theorem. Furthermore, we have \begin{equation*} \textbf{U}(t) - \textbf{V}(t) = \int_{s=0}^{1} \left({\it{\Phi}}^{t} \right)' \left(\textbf{V}_{0} + s \left(\textbf{U}_{0} - \textbf{V}_{0} \right) \right) \cdot \left(\textbf{U}_{0} - \textbf{V}_{0} \right)\!. \end{equation*} The result follows from Proposition 2.3. □ 3. A splitting scheme In this section, we split the Saut–Xu system (1.1) and give some estimates for the subproblems. We consider, separately, the transport part \begin{equation}\label{transport_part} \left\{ \begin{array}{@{}l@{}} \partial_{t} \zeta +\frac{\epsilon \sqrt{\mu}}{2} \left( \left({\mathcal{H}_{\mu}}^{2} + 1 \right) v \right) \partial_{x} \zeta = 0\\ \partial_{t} v + \frac{3 \epsilon \sqrt{\mu}}{2} v \partial_{x} v = 0 \end{array} \right. \end{equation} (3.1) and the dispersive part \begin{equation}\label{dispersive_part} \left\{ \begin{array}{@{}l@{}} \partial_{t} \zeta - {\mathcal{H}_{\mu}} v \epsilon \sqrt{\mu} \left( \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) \zeta \partial_{x} v - \frac{1}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} v \right) = \beta \sqrt{\mu} \partial_{x} \left(B_{\mu} v \right)\\ \partial_{t} v \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta - \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v = 0. \\ \end{array} \right. \end{equation} (3.2) We denote by $${\it{\Phi}}_{\mathcal{\rm A}}^{t}$$ the flow of System (3.1) and by $${\it{\Phi}}_{\mathcal{\rm D}}^{t}$$ the flow of System (3.2). Remark 3.1 Notice that we keep the term $$\zeta \partial_{x} v$$ in the first equation and we decompose $$v \partial_{x} \zeta$$ as $$v \partial_{x} \zeta = \partial_{x} \zeta \left({\mathcal{H}_{\mu}}^{2} +1 \right)v - \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} v$$. This will be useful for the local well-posedness of the dispersive part. In the following, we prove the local existence on large time for Systems (3.1) and (3.2). 3.1 The transport equation The system (3.1) is a transport equation. Then, it is easy to get the following result. Proposition 3.2 Let $$s_{1} \geq 0$$, $$s_{2} > \frac{3}{2}$$ and $$M > 0$$. We assume that $$\epsilon, \mu$$ satisfies Condition (1.3). Then, there exists a time $$T_{1} = T_{1} \left(M,\mu_{\max} \right) >0$$, such that if \begin{equation*} \left\lvert \zeta_{0} \right\rvert_{H^{s_{1}}} + \left\lvert v_{0} \right\rvert_{H^{s_{2}}} \leq M, \end{equation*} we have a unique solution $$\left(\zeta, v \right) \in \mathcal{C} \left(\left[0, \frac{T_{1}}{\epsilon} \right], H^{s_{1}}({\mathbb{R}}) \times H^{s_{2}}({\mathbb{R}}) \right)$$, to System (3.1) with initial data $$\left(\zeta_{0}, v_{0} \right)$$. Furthermore, we have, for all $$t\leq \frac{T_{1}}{\epsilon}$$, \begin{equation}\label{estim_int1} \left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}} \leq C(M,\mu_{\max}). \end{equation} (3.3) Finally, if $$s_{2} \geq 4$$ and $$M_{1} = \underset{0 \leq t \leq \frac{T_{1}}{\epsilon}}{\max} \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}-2}}$$ then for all $$t\leq \frac{T_{1}}{\epsilon}$$, we have \begin{equation}\label{borne1} \left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}} \leq e^{\epsilon C_{1} t} |{\mathbf{U}}_0|_{H^{s_1}\times H^{s_2}}, \end{equation} (3.4) where $$C_{1}>0$$ depends on $$M_{1}$$ and $$\mu_{\max}$$. Proof. The proof follows from the fact that the quasilinear system (3.1) is symmetric. Thanks to the Coifman–Meyer estimate (see Proposition A.3) in the Appendix, we get \begin{equation*} \frac{\rm d}{\rm dt} \left(\left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}}^{2} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}}^{2} \right) \leq C \epsilon \sqrt{\mu} \left( \left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}}^{2} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}}^{2} \right)^{\frac{3}{2}}\!. \end{equation*} Then, we see that the energy is bounded uniformly with respect to $$\epsilon$$ and $$\mu$$, and we get Estimate (3.3). For the second estimate, using the same trick that in Holden et al. (2013, Lemma 3.1), we notice that, if $$s_{1} \geq 4$$, \begin{equation*} \frac{\,{\rm d}}{{\rm d}t} \left(\left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}}^{2} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}}^{2} \right) \leq \epsilon \sqrt{\mu} \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}-2}} \left( \left\lvert \zeta(t,\cdot) \right\rvert_{H^{s_{1}}}^{2} + \left\lvert v(t,\cdot) \right\rvert_{H^{s_{2}}}^{2} \right)\!. \end{equation*} By applying the Gronwall lemma, we get the result. □ 3.2 The dispersive equation The system (3.2) contains all the dispersive terms of the Saut–Xu system. We have the following estimate for the flow. Proposition 3.3 Let $$N \geq 2$$ and $$b \in L^{\infty}({\mathbb{R}})$$. We assume that $$\epsilon, \beta, \mu$$ satisfy Condition (1.3). Then, there exists a time $$T_{2} = T_{2} \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right)$$, such that if \begin{equation*} \left\lvert \zeta_{0} \right\rvert_{H^{N+\frac{1}{2}}} + \left\lvert v_{0} \right\rvert_{H^{N}} + \left\lvert b \right\rvert_{L^{\infty}} \leq M, \end{equation*} we have a unique solution $$\left(\zeta,v \right) \in \mathcal{C} \left(\left[0, \frac{T_{2}}{\delta} \right], H^{N+\frac{1}{2}} ({\mathbb{R}}) \times H^{N}({\mathbb{R}}) \right)$$ to the system (3.2) with initial data $$\left(\zeta_{0},v_{0} \right)$$. Furthermore, we have, for all $$t \leq \frac{T_{2}}{\delta}$$, \begin{equation}\label{estim_int2} \left\lvert \zeta(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}}} + \left\lvert v(t,\cdot) \right\rvert_{H^{N}} \leq C \left( M,\mu_{\max}, \frac{1}{\mu_{\min}} \right)\!. \end{equation} (3.5) Finally, if $$N \geq 7$$, and \begin{equation*} M_{1} = \underset{0 \leq t \leq \frac{T_{2}}{\delta}}{\max} \left( \left\lvert \zeta(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}-2}} + \left\lvert v(t,\cdot) \right\rvert_{H^{N-2}} \right)\!, \end{equation*} then for all $$t\leq \frac{T_{2}}{\delta}$$, we have \begin{equation}\label{borne2} \left\lvert \zeta(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}}} + \left\lvert v(t,\cdot) \right\rvert_{H^{N}} \leq e^{\delta C_{2} t} |{\mathbf{U}}_0|_{H^{N+1/2} \times H^N }, \end{equation} (3.6) where $$C_{2}$$ is a positive constant that depends on $$\mu_{\max}, \frac{1}{\mu_{\min}}, M_{1}$$. Proof. The proof is an adaptation of the proof of Theorem 2.2 and Saut & Xu (2012, Part IV). We notice that, in the proof of Saut and Xu, the transport part can be treated separately and does not influence the control of the other terms. Hence, we can use the same symmetrizer $$\mathcal{S}$$ that in Theorem 2.2 (see (2.6)) and we get \begin{equation*} \frac{{\rm d}}{{\rm d}t} \mathcal{E}^{N} \! \left(\zeta,v \right) \leq C \left(\frac{1}{\mu_{\min}} \right) \left(\frac{\epsilon}{\nu} \mathcal{E}^{N} \! \left(\zeta,v \right)^{\frac{3}{2}} + \frac{\beta}{\nu} \mathcal{E}^{N} \! \left(\zeta,v \right) \right)\!. \end{equation*} Then, by Remark 2.1, we get Estimate (3.5). Furthermore, we notice that we use the same trick as in Holden et al. (2013, Lemma 3.1). By keeping the same notations as in Theorem 2.2, we get from equations (2.7) and (2.8) that \begin{equation*} \frac{{\rm d}}{{\rm d}t} \mathcal{E}^{N} \! \left(\textbf{U} \right) \leq \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max} \right) \left( \left\lvert \tilde{\mathcal{G}}^{N} \right\rvert_{ H^{1/2}\times L^{2}} + \sqrt{\mathcal{E}^{2} \left(\textbf{U} \right)} + \left\lvert b \right\rvert_{\infty} \right) \mathcal{E}^{N} \! \left(\textbf{U} \right)\!, \end{equation*} where $$\tilde{\mathcal{G}}^{N} = (\tilde{\mathcal{G}}^{N}_{1}, \tilde{\mathcal{G}}^{N}_{2})^{t}$$ with \begin{equation*} \begin{aligned} &\tilde{\mathcal{G}}^{N}_{1} = - \partial_{x}^{N} \left( [{\mathcal{H}_{\mu}}, \zeta] {\mathcal{H}_{\mu}} \partial_{x} v + \zeta ({\mathcal{H}_{\mu}}^{2}+1) \partial_{x} v \right) - \frac{1}{2} \underset{1 \leq \gamma \leq N-1}{\sum} C^{\gamma}_{N} {\mathcal{H}_{\mu}} ( \partial_{x}^{\gamma} v {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta) - \frac{1}{2} \partial_{x} \zeta ({\mathcal{H}_{\mu}}^{2}+1) \partial_{x}^{N} v\\ &\tilde{\mathcal{G}}^{N}_{2} = \frac{1}{2} \underset{1 \leq \gamma \leq N-1}{\sum} C^{\gamma}_{N} \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta + \frac{1}{2} \underset{1 \leq \gamma \leq N}{\sum} C^{\gamma}_{N} \partial_{x}^{\gamma} v {\mathcal{H}_{\mu}}^{2} \partial_{x}^{1+N-\gamma} v. \end{aligned} \end{equation*} To explain how we can adapt the trick used in Holden et al. (2013, Lemma 3.1), we focus our attention to one term. For $$1 \leq \gamma \leq N-1$$, we have to control $$\left\lvert \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{L^2}$$. If $$\gamma \leq \lfloor \frac{N}{2} \rfloor$$, we get from Propositions A.4 and A.1 in the Appendix that \begin{equation*} \left\lvert \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{L^2} \leq \left\lvert \partial_{x}^{1+\gamma} \zeta \right\rvert_{H^{1}} \left\lvert {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{L^2} \leq C \left(\mu_{\max} \right) \left\lvert \zeta \right\rvert_{H^{2 + \lfloor \frac{N}{2} \rfloor}} \left\lvert \zeta \right\rvert_{H^{N}}\!, \end{equation*} whereas if $$\gamma > \lfloor \frac{N}{2} \rfloor$$, we have \begin{equation*} \left\lvert \partial_{x}^{1+\gamma} \zeta {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{L^2} \leq \left\lvert \partial_{x}^{1+\gamma} \zeta \right\rvert_{L^2} \left\lvert {\mathcal{H}_{\mu}} \partial_{x}^{1+N-\gamma} \zeta \right\rvert_{H^1} \leq C \left(\mu_{\max} \right) \left\lvert \zeta \right\rvert_{H^{N}} \left\lvert \zeta \right\rvert_{H^{2+ \lfloor \frac{N}{2} \rfloor}}\!. \end{equation*} We can mimic this method to control the other terms of $$\tilde{\mathcal{G}}^{N}$$ and, thanks to Propositions A.1, A.2 and A.4 in the Appendix, we obtain if $$N \geq 7$$ that \begin{equation*} \frac{{\rm d}}{{\rm d}t} \mathcal{E}^{N} \! \left(\zeta,v \right) \leq \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max} \right) \left( \left\lvert \zeta(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}-2}} + \left\lvert v(t,\cdot) \right\rvert_{H^{N-2}} + \left\lvert b \right\rvert_{L^{\infty}} \right) \mathcal{E}^{N} \! \left(\zeta,v \right)\!. \end{equation*} Then, Estimate (3.6) follows. □ Remark 3.4 Under the assumption of Proposition 3.3 and if $$N \geq 7$$, we get from relations (3.4) and (3.6) that, there exists a time $$T_{3}>0$$, such that for all $$t\in \left[0, \frac{T_{3}}{\delta} \right]$$, \begin{equation*} |\mathcal{Y}^t {\mathbf{U}}_0 |_{ H^{N+1/2}\times H^{N}} \leq e^{C_{3} \delta t} |{\mathbf{U}}_0|_{H^{N+1/2}\times H^{N}}, \end{equation*} where $$C_{3} = C \left(\left\lvert {\mathbf{U}}_0 \right\rvert_{H^{N+1/2-2}\times H^{N-2}}, \mu_{\max}, \frac{1}{\mu_{\min}} \right)$$ and $$T_{3} = C \left( \left\lvert {\mathbf{U}}_0 \right\rvert_{H^{N+1/2}\times H^{N}}, \mu_{\max}, \frac{1}{\mu_{\min}} \right)$$. 4. Error estimates The goal of this part is to prove the main result of this article (Theorem 4.6). Our analysis is based on energy estimates. 4.1 The local error estimate The local error is the following quantity \begin{equation}\label{localerror} e \left(t, \textbf{U}_{0} \right) = {\it{\Phi}}^{t} \textbf{U}_{0} - \mathcal{Y}^{t} \textbf{U}_{0}. \end{equation} (4.1) Our approach is similar to the one developed in Chartier et al. (2016). We use the fact that $${\it{\Phi}}^{t} \textbf{U}_{0}$$ satisfies a symmetrizable system. Therefore, $$e$$ satisfies this system up to a remainder and then, we can control $$e$$, thanks to energy estimates. In the following, we give different technical lemmas, to control the local error. We recall that the transport operator is the operator $$\mathcal{A}$$ \begin{equation*} \mathcal{A} \left(\zeta, v \right) = - \frac{\epsilon \sqrt{\mu}}{2} \begin{pmatrix} \left(\left({\mathcal{H}_{\mu}}^{2} + 1 \right) \! v \right) \partial_{x} \zeta \\ 3 v \partial_{x} v \end{pmatrix}\!. \end{equation*} The following proposition gives an estimate of the differential of the transport operator. Lemma 4.1 Let $$s_{1}, s_{2} \geq 0$$ and $$\epsilon, \mu$$ satisfying Condition (1.3). Then, \begin{equation*} \left\lvert \mathcal{A}' (\zeta,v).(\eta,w) \right\rvert_{H^{s_{1}} \times H^{s_{2}}} \leq \epsilon C(\mu_{\max}) \left\lvert (\zeta,v) \right\rvert_{H^{s_{1}+1} \times H^{s_{2}+1}} \left\lvert (\eta,w) \right\rvert_{H^{s_{1}+1} \times H^{s_{2}+1}}\!. \end{equation*} Proof. We have \begin{equation*} \mathcal{A}' (\zeta,v).(\eta,w) = - \frac{\epsilon \sqrt{\mu}}{2} \begin{pmatrix} \left( \left({\mathcal{H}_{\mu}}^{2} + 1 \right) \! v \right) \partial_{x} \eta + \left( \left({\mathcal{H}_{\mu}}^{2} + 1 \right) \! w \right)\partial_{x} \zeta \\ 3 w \partial_{x} v+ 3 v \partial_{x} w \end{pmatrix}\!, \end{equation*} and the estimate follows from Propositions A.1 and A.4 in the Appendix. □ We can do the same for the dispersive part (using also Proposition A.1 in the Appendix). We recall that the dispersive operator is the operator $$\mathcal{D}$$ \begin{equation*} \mathcal{D}(\zeta,v) = \begin{pmatrix} {\mathcal{H}_{\mu}} v \epsilon \sqrt{\mu} \left( \frac{1}{2} {\mathcal{H}_{\mu}} \left(v \partial_{x} {\mathcal{H}_{\mu}} \zeta \right) + {\mathcal{H}_{\mu}} \left( \zeta \partial_{x} {\mathcal{H}_{\mu}} v \right) + \zeta \partial_{x} v - \frac{1}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} v \right) - \beta \sqrt{\mu} \partial_{x} \left(B_{\mu} v \right) \\ - \partial_{x} \zeta + \frac{\epsilon \sqrt{\mu}}{2} \partial_{x} \zeta {\mathcal{H}_{\mu}} \partial_{x} \zeta + \frac{\epsilon \sqrt{\mu}}{2} v {\mathcal{H}_{\mu}}^{2} \partial_{x} v \end{pmatrix}\!. \end{equation*} Lemma 4.2 Let $$s >0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Then, \begin{equation*} \left\lvert \mathcal{D}' (\zeta,v).(\eta,w) \right\rvert_{H^{s} \times H^{s}} \leq C(\mu_{\max}) \left(1 \beta \left\lvert b \right\rvert_{L^{\infty}} \epsilon \left\lvert (\zeta,v) \right\rvert_{H^{s+1} \times H^{s+1}} \right) \left\lvert (\eta,w) \right\rvert_{H^{s+1} \times H^{s+1}}\!. \end{equation*} Furthermore, we have to control the derivative of the flow $${\it{\Phi}}_{\mathcal{A}}^{t}$$ with respect to the initial data. We denote it by $$\left({\it{\Phi}}_{\mathcal{A}}^{t} \right)'$$. Lemma 4.3 Let $$s_{1}, s_{2} \geq 0$$, $$M > 0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Let $$\left( \zeta_{0}, v_{0} \right) \in H^{s_{1}+1} \times H^{s_{2}+1} ({\mathbb{R}^{d}})$$ such that, \begin{equation*} \left\lvert \left( \zeta_{0}, v_{0} \right) \right\rvert _{H^{s_{1}+1} \times H^{s_{2}+1}} \leq M. \end{equation*} Then, there exists a time $$T = T(M,\mu_{\max})$$, such that $$\left({\it{\Phi}}_{\mathcal{A}}^{t} \right)' \left( \zeta_{0}, v_{0} \right) \cdot \left(\eta_{0},w_{0} \right)$$ exists for all $$t \in \left[0, \frac{T}{\delta} \right]$$, and if we denote \begin{equation*} \begin{pmatrix} \eta \\ w \end{pmatrix} = \left({\it{\Phi}}_{\mathcal{A}}^{t} \right)' \left( \zeta_{0}, v_{0} \right) \cdot \left(\eta_{0},w_{0} \right) \end{equation*} for all $$0 \leq t \leq \frac{T}{\delta }$$, \begin{equation*} \left\lvert \left( \eta, w \right)(t,\cdot) \right\rvert_{H^{s_{1}} \times H^{s_{2}}} \leq \left\lvert \left( \eta_{0}, w_{0} \right) \right\rvert _{H^{s_{1}} \times H^{s_{2}}} C \left(\mu_{\max}, M \right)\!. \end{equation*} Proof. The quantity $$\left( \eta, w \right)$$ satisfies the following linear system \begin{equation*} \left\{ \begin{array}{@{}l@{}} \partial_{t} \eta + \frac{\epsilon \sqrt{\mu}}{2} \left({\mathcal{H}_{\mu}}^{2} + 1 \right) v \partial_{x} \eta + \frac{\epsilon \sqrt{\mu}}{2} \left({\mathcal{H}_{\mu}}^{2} + 1 \right) w \partial_{x} \zeta = 0,\\ \partial_{t} w + \frac{3 \epsilon \sqrt{\mu}}{2} v \partial_{x} w + \frac{3 \epsilon \sqrt{\mu}}{2} w \partial_{x} v = 0, \end{array} \right. \end{equation*} where $$\left( \zeta, v \right) = {\it{\Phi}}_{\mathcal{A}}^{t} \left( \zeta_{0}, v_{0} \right)$$. The result follows from energy estimates, the Gronwall lemma and Proposition 3.2. □ In the following, we use the fact $${\it{\Phi}}^{t}_{\mathcal{A}} \circ {\it{\Phi}}^{t}_{\mathcal{D}}$$ satisfies the Saut–Xu system (1.1) up to a remainder. The following lemma is the key point for the control of this remainder. Lemma 4.4 Let $$N \geq 2$$, $$M > 0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Let $$\textbf{U} = \left( \zeta, v \right) \in H^{N+\frac{1}{2}} \times H^{N} ({\mathbb{R}^{d}})$$ such that, \begin{equation*} \left\lvert b \right\rvert_{L^{\infty}} + \left\lvert\textbf{U} \right\rvert _{H^{N+\frac{1}{2}} \times H^{N} ({\mathbb{R}})} \leq M. \end{equation*} Then, there exists a time $$T = T \left(M, \mu_{\max}, \frac{1}{\mu_{\min}} \right) > 0$$, such that $${\it{\Phi}}^{t}_{\mathcal{A}} \left( \textbf{U} \right)$$ exists for all $$0 \leq t \leq \frac{T}{\delta}$$, and furthermore, \begin{equation*} \left\lvert \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left(\textbf{U} \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left(\textbf{U} \right) \right) \right\rvert_{H^{N-2} \times H^{N-2}} \leq \epsilon C \left(M, \mu_{\max}, \frac{1}{\mu_{\min}} \right) t. \end{equation*} Proof. The existence of $$T$$ follows from Proposition 3.2. Then, we notice that \begin{equation*} \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left(\textbf{U} \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left(\textbf{U} \right) \right) = \int_{0}^{t} \mathcal{A}' \left({\it{\Phi}}_{\mathcal{A}}^{s} \left(\textbf{U} \right) \right) \cdot \left( \left({\it{\Phi}}^{s}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left( \textbf{U} \right) \right) - \mathcal{D}' \left( {\it{\Phi}}_{\mathcal{A}}^{s} \left(\textbf{U} \right) \right) \cdot \mathcal{A} \left( {\it{\Phi}}^{s}_{\mathcal{A}} \left( \textbf{U} \right) \right)\!. \end{equation*} Using Lemmas 4.1, 4.2 and Proposition 3.2, we get \begin{align*} \left\lvert \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left(\textbf{U} \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left(\textbf{U} \right) \right) \right\rvert_{H^{N-2} \times H^{N-2}} &\leq C \left( \mu_{\max}, M \right) \int_{0}^{t} \epsilon \left\lvert \left({\it{\Phi}}^{s}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left( \textbf{U} \right) \right\rvert_{H^{N-1} \times H^{N-1}}\\ &\quad{} + \left\lvert \mathcal{A} \left( {\it{\Phi}}^{s}_{\mathcal{A}} \left( \textbf{U} \right) \right) \right\rvert_{H^{N-1} \times H^{N-1}}\!. \end{align*} Then, using Lemma 4.3, the product estimate A.4 and the expression of $$\mathcal{A}$$, we obtain \begin{equation*} \left\lvert \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left(\textbf{U} \right) \cdot \mathcal{D} \left(\textbf{U} \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left(\textbf{U} \right) \right) \right\rvert_{H^{N-2} \times H^{N-2}} \leq \epsilon C \left( \mu_{\max}, M \right) \int_{0}^{t} \left\lvert \mathcal{D} \left( \textbf{U} \right) \right\rvert_{H^{N-1} \times H^{N-1}} + \left\lvert {\it{\Phi}}^{s}_{\mathcal{A}} \left( \textbf{U} \right) \right\rvert^{2}_{H^{N} \times H^{N}}\!. \end{equation*} Finally, the result follows from the expression of $$\mathcal{D}$$, the product estimate A.4 and Proposition A.1 in the Appendix. □ We can now give the main result of this part, the local error estimate. Proposition 4.5 Let $$N \geq 4$$, $$M > 0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Let $$\textbf{U}_{0} = \left( \zeta_{0}, v_{0} \right)$$, such that \begin{equation*} \left\lvert b \right\rvert_{L^{\infty}} + \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq M. \end{equation*} Then, there exists a time $$T_{4} = T_{4} \left(M, \frac{1}{\mu_{\min}}, \mu_{\max} \right) > 0$$, such that the local error $$e \left(t, \textbf{U} \right)$$ defined in (4.1) exists for all $$0 \leq t \leq \frac{T_{4}}{\delta }$$, and furthermore, \begin{equation*} \left\lvert e \left(t, \textbf{U}_{0} \right) \right\rvert_{H^{N-4 + \frac{1}{2}} \times H^{N-4}} \leq \delta C_{4} t^{2}, \end{equation*} where $$C_{4} = C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right)$$. Proof. From Propositions 3.2 and 3.3, we obtain the existence of $$T$$. We denote \begin{equation*} \textbf{U}(t) = \begin{pmatrix} \zeta(t) \\ v(t) \end{pmatrix} = {\it{\Phi}}^{t} \left(\textbf{U}_{0} \right) \text{ and }\textbf{V}(t) = \begin{pmatrix} \eta(t) \\ w(t) \end{pmatrix} = {\it{\Phi}}^{t}_{\mathcal{A}} \left( {\it{\Phi}}^{t}_{\mathcal{D}} \left(\textbf{U}_{0} \right) \right)\!. \end{equation*} Then, from Theorem 2.2 and Propositions 3.2 and 3.3, we also have, for all $$0 \leq t \leq \frac{T}{\delta}$$, \begin{equation}\label{controlUV} \left\lvert \textbf{U}(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} + \left\lvert \textbf{V}(t, \cdot) \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right)\!. \end{equation} (4.2) We know that $$\left(\zeta, v \right)$$ satisfies the Saut–Xu system (1.1). Furthermore, $$\left(\eta, w \right)$$ also satisfies the Saut–Xu system (1.1) up to a remainder \begin{equation*} \partial_{t} \begin{pmatrix} \eta \\ w \end{pmatrix} = \mathcal{A} \left(\eta,w\right) + \mathcal{D} \left(\eta,w\right) + \mathcal{R}(t), \end{equation*} where $$\mathcal{R}(t) = \left({\it{\Phi}}^{t}_{\mathcal{A}} \right)' \left({\it{\Phi}}^{t}_{\mathcal{D}} \left(\textbf{U}_{0} \right) \right) \cdot \mathcal{D} \left({\it{\Phi}}^{t}_{\mathcal{D}} \left(\textbf{U}_{0} \right) \right) - \mathcal{D} \left( {\it{\Phi}}_{\mathcal{A}}^{t} \left({\it{\Phi}}^{t}_{\mathcal{D}} \left(\textbf{U}_{0} \right) \right) \right)$$. Therefore, the local error $$e$$ satisfies the following system \begin{equation}\label{eq_on_e} \partial_{t} e = \begin{pmatrix} 0 & H_{\mu} \\- \partial_{x} & 0 \end{pmatrix} e + \begin{pmatrix} 0 & \beta \sqrt{\mu} B_{\mu} \\ 0 & 0 \end{pmatrix} e + \mathcal{T}_{\mu} \left(\left(\zeta,v \right), \left(\eta,w \right)\right) - \mathcal{R}(t), \end{equation} (4.3) where the operator $$\mathcal{T}_{\mu} \left(\textbf{U}, \textbf{V} \right)$$ is quadratic and satisfies the following estimate, for $$0 \leq s \leq N-1$$, \begin{equation}\label{controlnonlin} \left\lvert \mathcal{T}_{\mu} \left(\left(\zeta,v \right), \left(\eta,w \right)\right) \right\rvert_{H^{s} \times H^{s}} \leq \epsilon C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \left\lvert e \right\rvert_{H^{s+1} \times H^{s+1}}\!. \end{equation} (4.4) Then, since $$e_{|t=0}=0$$, \begin{equation*} e(t,\cdot) = \int_{0}^{t} \partial_{t} e(s,\cdot)\,{\rm d}s, \end{equation*} and since $$e$$ satisfies (4.3), we obtain, thanks to Estimates (4.2) and (4.4) and Lemma 4.4, \begin{equation}\label{firstenergyest} \left\lvert e \left(t, \cdot \right) \right\rvert_{H^{N-2} \times H^{N-2}} \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) t. \end{equation} (4.5) Furthermore, we recall that the Saut–Xu system (1.1) is symmetrizable, thanks to the symmetrizer (see Theorem 2.2) \begin{equation*} \mathcal{S} = \begin{pmatrix} \frac{D}{\tanh(\sqrt{\mu} D)} & 0 \\ 0 & 1 \end{pmatrix}\!. \end{equation*} Therefore, applying $$\mathcal{S}$$ to the system (4.3), and using the fact that $$\sqrt{\left( \mathcal{S} \cdot, \cdot \right)}$$ is a norm equivalent to the $$H^{\frac{1}{2}} \times L^{2}$$-norm, we obtain, thanks to Estimates (4.2), (4.4) and (4.5) and Lemma 4.4, \begin{equation*} \frac{\rm d}{\rm dt} \mathcal{F}(e) \leq C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \left(\beta \mathcal{F}(e) + \epsilon t \sqrt{\mathcal{F}(e)} \right)\!, \end{equation*} where $$\mathcal{F}(e) = \sum \limits_{|\alpha| \leq N-4} \left(S \partial_{x}^{\alpha} e, \partial_{x}^{\alpha} e \right)$$. Then, we get \begin{equation*} \mathcal{F}(e)(t) \leq \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \int_{0}^{t} \mathcal{F}(e)(s) + s \sqrt{\mathcal{F}(e)(s)} \,{\rm d}s. \end{equation*} Denoting $$\mathcal{M}(t) = \underset{[0,t]}{\max} \sqrt{\mathcal{F}(e)(t)}$$, we have \begin{equation*} \mathcal{M}(t) \leq \delta C \left(\frac{1}{\mu_{\min}}, \mu_{\max}, M \right) \int_{0}^{t} \mathcal{M}(s) + s \,{\rm d}s, \end{equation*} and the result follows from the Grönwall’s lemma. □ 4.2 Global error estimate In this part, we prove our main result. We denote by \begin{equation*} \textbf{U}_{k} = \left(\mathcal{Y}^{{\it{\Delta}} t}\right)^k \textbf{U}_{0} \end{equation*} the approximate solution and by $$ {\mathbf{U}}(t_k) := {\it{\Phi}}^{k {\it{\Delta}} t} {\mathbf{U}}_0$$ the exact solution at the time $$t_k=k {\it{\Delta}} t$$. Theorem 4.6 Let $$N \geq 7$$, $$M > 0$$, $$\epsilon,\beta, \mu$$ satisfying Condition (1.3) and $$b \in L^{\infty}({\mathbb{R}})$$. Let $$\textbf{U}_{0} = \left( \zeta_{0}, v_{0} \right)$$, such that \begin{equation*} \left\lvert b \right\rvert_{L^{\infty}} + \left\lvert \textbf{U}_{0} \right\rvert_{H^{N+\frac{1}{2}} \times H^{N}} \leq M. \end{equation*} Let $$\textbf{U}_{0}(t, \cdot)$$ the solution of the Saut–Xu equations (1.1) with initial data $$\textbf{U}_{0}$$ defined on $$\left[0, \frac{T}{\delta} \right]$$. Then, there exist constants $$A\gamma, \nu, {\it{\Delta}} t_0, C_0>0$$, such that for all $${\it{\Delta}} t \in ]0,{\it{\Delta}} t_0]$$ and for all $$n \in {\mathbb{N}}$$, such that $$0 \leq n {\it{\Delta}} t \leq \frac{T}{\delta }$$, \begin{equation*} | {\mathbf{U}}_n|_{H^{N + \frac{1}{2}} \times H^{N}} \leq \nu \text{ and } \left\lvert {\it{\Phi}}^{n {\it{\Delta}} t} \left( \textbf{U}_{0} \right) - \left( \mathcal{Y}^{{\it{\Delta}} t} \right)^{n} \left( \textbf{U}_{0} \right) \right\rvert_{H^{N-4 + \frac{1}{2}} \times H^{N-4}} \leq \gamma {\it{\Delta}} t. \end{equation*} Proof. The proof is based on a Lady’s Windermere’s fan argument and is similar to the one in Carles (2013) (see also Holden et al., 2013). To simplify the notations, we forget the dependence on $$\frac{1}{\mu_{\min}}$$ and $$\mu_{\max}$$ in all the constants. We denote by $$X^{N}$$ the following space \begin{equation*} X^{N} = H^{N + \frac{1}{2}} \times H^{N}. \end{equation*} Thanks to Theorem 2.2, there exists $$\rho$$ such that, for all $${t^k = k {\it{\Delta}} t } \in \left[ 0,\frac{T}{\delta } \right]$$, \begin{equation*} \left\lvert {\mathbf{U}}(t_k) \right\rvert_{X^{N}} = \left\lvert {\it{\Phi}}^{t_k} \textbf{U}_{0} \right\rvert_{X^{N}} \leq \rho. \end{equation*} We prove by induction that there exists $${\it{\Delta}} t_0, \gamma, \nu$$, such that if $$0 < {\it{\Delta}} t \leq {\it{\Delta}} t_0,$$ for all $$n \in {\mathbb{N}}$$ with $$n {\it{\Delta}} t \leq \frac{T}{\delta}$$, \begin{align*} &(i) \text{ } |{\mathbf{U}}_n-{\mathbf{U}}(t_n) |_{X^{N-4}} \leq \gamma {\it{\Delta}} t,\\ &(ii) \text{ } |{\mathbf{U}}_n|_{X^{N}} \leq e^{C_{3}(M_{1}) \delta n {\it{\Delta}} t} \left\lvert \textbf{U}_{0} \right\rvert_{X^{N}} \leq M_{0},\\ &(iii) \text{ } |{\mathbf{U}}_n|_{X^{N-2}} \leq M_{1},\\ &(iv) \text{ } |{\mathbf{U}}_n|_{X^{N-4}} \leq 2 \rho, \end{align*} with \begin{align*} &M_{1} = e^{C_{3}(2 \rho) T} M, M_{0} = e^{C_{3}(M_{1}) T} M, \gamma = T \max(K,1) C_{4} \left(e^{C_{0}(M_{0}) T} M_{0} \right)\!,\\ &{\it{\Delta}} t_{0} = \min \left(T,T_{0}(M_{0}), T_{3}(M_{0}), T_{4}(M_{0}), \frac{\rho}{\gamma} \right),K=K \left( M_{0} e^{T C_{0}(M_{0})} \right)\!, \end{align*} where $$C_{0}$$, $$T_{0}$$, $$C_{3}$$, $$T_{3}$$, $$C_{4}$$, $$T_{4}$$ and $$K$$ are constants from Theorem 2.2, Remark 3.4, Proposition 4.5 and Inequality (2.11). The above properties are satisfied for $$n=0$$. Let $$n\geq 1$$, and suppose that the induction assumptions are true for $$0 \leq k \leq n-1$$. First, we have the following telescopic series (see Carles, 2013; Holden et al., 2013) \begin{equation}\label{telecsopic_series} \textbf{U}_{n} - {\mathbf{U}}(t_n) = \underset{0 \leq k \leq n-1}{\sum} {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} \mathcal{Y}^{{\it{\Delta}} t} \textbf{U}_{k} - {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} {\it{\Phi}}^{{\it{\Delta}} t} \textbf{U}_{k}. \end{equation} (4.6) For $$k\leq n-2$$, since $$\mathcal{Y}^{{\it{\Delta}} t} {\mathbf{U}}_k = {\mathbf{U}}_{k+1}$$, using the induction assumption (ii), we have \begin{equation*} |\mathcal{Y}^{{\it{\Delta}} t} {\mathbf{U}}_k|_{X^{N-3}} \leq M_{0}, \end{equation*} and from Theorem 2.2, we get \begin{equation*} |{\it{\Phi}}^{{\it{\Delta}} t} {\mathbf{U}}_k|_{X^{N-3}} \leq e^{C_{0}(M_{0}) T} M_{0}. \end{equation*} Therefore, from Proposition 2.4 and up to replacing $$K=K \left( M_{0}\, e^{T C_{0}(M_{0})} \right)$$ with $$\max(K,1)$$, we obtain, for $$k \leq n-1$$ and $$n {\it{\Delta}} t \leq \frac{T}{\delta}$$, \begin{equation*} \left\lvert {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} \mathcal{Y}^{{\it{\Delta}} t} \textbf{U}_{k} - {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} {\it{\Phi}}^{{\it{\Delta}} t} \textbf{U}_{k} \right\rvert_{X^{N-4}} \leq K \left\lvert \mathcal{Y}^{{\it{\Delta}} t} {\mathbf{U}}_k- {\it{\Phi}}^{{\it{\Delta}} t} {\mathbf{U}}(t_k) \right\rvert_{X^{N-4}}\!. \end{equation*} Then, using Proposition 4.5 and Inequality (ii), we infer \begin{equation*} \left\lvert {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} \mathcal{Y}^{{\it{\Delta}} t} \textbf{U}_{k} - {\it{\Phi}}^{(n-k-1) {\it{\Delta}} t} {\it{\Phi}}^{{\it{\Delta}} t} \textbf{U}_{k} \right\rvert_{X^{N-4}} \leq \delta C_{4} \left( e^{C_{0}(M_{0}) T} M_{0} \right) K ({\it{\Delta}} t)^2. \end{equation*} Therefore, using the telescopic series (4.6), we get \begin{equation*} |\textbf{U}_{n} - {\mathbf{U}}(t_n)|_{X^{N-4}} \leq n C_{4} \left( e^{C_{0}(M_{0}) T} M_{0} \right) K \delta ({\it{\Delta}} t)^2 \leq C_{4} \left( e^{C_{0}(M_{0}) T} M_{0} \right) K T {\it{\Delta}} t. \end{equation*} For Estimate (ii), using Remark 3.4 and the induction assumptions (iii) and (ii), we have \begin{equation*} \left\lvert \textbf{U}_{n} \right\rvert_{X^{N}} = \left\lvert \mathcal{Y}^{{\it{\Delta}} t} \left( \textbf{U}_{n-1} \right) \right\rvert_{X^{N}} \leq e^{\delta C_{3}(M_{1}) {\it{\Delta}} t} \left\lvert \textbf{U}_{n-1} \right\rvert_{X^{N}} \leq e^{C_{3}(M_{1}) \delta n {\it{\Delta}} t} \left\lvert \textbf{U}_{0} \right\rvert_{X^{N}} \leq M_{0}. \end{equation*} We get Estimate (iii) in the same way, using the induction assumptions (iv) and (iii). Finally, for Estimate (iv), using (i), we have \begin{align*} \left\lvert \textbf{U}_{n} \right\rvert_{X^{N-4}} \leq \left\lvert \textbf{U}_{n} - \textbf{U}(t_{n}) \right\rvert_{X^{N-4}} + \left\lvert \textbf{U}(t_{n}) \right\rvert_{X^{N-4}} \leq \gamma {\it{\Delta}} t \rho \leq 2 \rho. \end{align*} □ 5. Numerical experiments The aim of this section is to numerically verify the Lie method convergence rate in $$\mathcal{O}({\it{\Delta}} t)$$ for the Saut–Xu system (1.1) and to illustrate some physical phenomena. In other works and particularly on the whole water waves problem (see, e.g., Craig & Sulem, 1993; Nicholls & Reitich, 2001; Guyenne & Nicholls, 2007–08; and references therein), several authors use a discrete Fourier transform even for the transport part. They observe spurious oscillations in the wave profile that lead to instabilities. These errors seem to appear when they evaluate the nonlinear part via Fourier transform, because additional terms appear in the approximation, which is the aliasing phenomenon. To fix this problem, they apply at every time step a low-pass filter. The main interest of our scheme is that we do not need one, because we use a finite difference method to approximate the nonlinear part. For the dispersive equation (3.2), we use the forward Euler discretization in time and for the spatial discretization, we consider the FFT implemented in Matlab. In this scheme, the interval $$[0,1]$$ is discretized by $$N$$ equidistant points, with spacing $${\it{\Delta}} x = 1/N$$. The spatial grid points are then given by $$x_j = j/N$$, $$j=0,...,N$$. Therefore, if $$u_j(t)$$ denotes the approximate solution to $$u(t,x_j)$$, the discrete Fourier transform of the sequence $$ \left\lbrace u_j \right\rbrace_{j = 0}^{N-1} $$ is defined by \begin{equation*} \hat{u}(k) = \mathcal{F}^d_k (u_j) = \sum_{j=0}^{N-1} u_j\, e^{-2i \pi j k /N }, \end{equation*} for $$k = 0,\cdots, N-1$$, and the inverse discrete Fourier transform is given by \begin{equation*} u_j = \mathcal{F}_{j}^{-d}( \hat{u}_k ) = \frac{1}{N} \sum_{k = 0}^{N -1} \hat{u}_k \,e^{ 2 i \pi k x_j}, \end{equation*} for $$j = 0, \cdots,N-1 $$. Here, $$\mathcal{F}^d$$ denotes the discrete Fourier transform and $$\mathcal{F}^{-d}$$ its inverse. Then, in what follows, the numerical scheme to solve (3.2) is given by \begin{equation} \begin{pmatrix} \zeta^{n+1}_j \\ v_j^{n+1} \end{pmatrix} = \begin{pmatrix} \zeta^{n}_j \\ v_j^{n} \end{pmatrix} - {\it{\Delta}} t \begin{pmatrix} F_j^{n} + S_j^n \\ G_j^{n} \end{pmatrix}\!, \label{subeq1} \end{equation} (5.1) where $$S_j^n = \beta \sqrt{\mu} \mathcal{F}^{-d}_j(i k \mathcal{F}^d_k (B_\mu v^n_j ))$$ and $$F_j^n = I_1 + I_2 $$ with \begin{align*} I_1 &= \mathcal{F}^{-d}_j \left( i \, \tanh\left(\sqrt{\mu} k\right) \left( -1 + \frac{\epsilon \sqrt{\mu}}{2} \mathcal{F}^d_k\left( v^n_j \mathcal{F}^{-d}_j\left(k \tanh\left(\sqrt{\mu k} \right) \hat{\zeta}^n_k \right) \right) \right.\right.\\ &\left.\left.+ \epsilon \sqrt{\mu} \mathcal{F}^d_k\left( \zeta^n_j \mathcal{F}^{-d}_j\left(k \tanh(\sqrt{\mu k} ) \hat{v}^n_k \right) \right) \right) \right) \end{align*} \begin{equation*} I_2 = \zeta_j^n \mathcal{F}^{-d}_j\left( i k \hat{v}^n_k \right) + \frac{1}{2} \mathcal{F}^{-d} \left( i k \hat{\zeta}^n_k \right) \mathcal{F}_j^{-d} \left( \tanh(\sqrt{\mu} k )^2 \hat{v}^n_k \right)\!. \end{equation*} To approximate equation (3.1), we use the following finite difference scheme \begin{equation} \begin{pmatrix} \zeta^{n+1}_j \\ v_j^{n+1} \end{pmatrix} = \begin{pmatrix} \zeta^{n}_j \\ v_j^{n} \end{pmatrix} - {\it{\Delta}} t \frac{\epsilon \sqrt{\mu} }{2} \begin{pmatrix} G^n_1 \\ 3 G^n_2 \end{pmatrix}\!, \label{subeq2} \end{equation} (5.2) where \begin{equation*} G_1 = w_j^n \frac{\zeta^n_{j+1}-\zeta^n_{j-1}}{2 {\it{\Delta}} x} - \frac{{\it{\Delta}} t}{2 {\it{\Delta}} x^2} (w^n_j)^2 \left( \zeta^n_{j-1}-2\zeta^n_j+\zeta^n_{j+1} \right) \end{equation*} with $$w_j^n = - \mathcal{F}^{-d}_j \left( \tanh( \sqrt{\mu} k)^2 \hat{v}^n_k \right) + v_j^n$$ and $$ G_2^n = \frac{(v^n_{j+1})^2 - (v^n_{j-1})^2 }{2 {\it{\Delta}} x} - \frac{{\it{\Delta}} t}{2 {\it{\Delta}} x^2} \left( v^n_{j+1/2} \left( (v^n_{j+1})^2 - (v^n_{j})^2 \right) - v^n_{j-1/2} \left( (v^n_j)^2 - (v^n_{j-1} )^2 \right) \right) $$ with $$v^n_{j \pm 1/2} = \frac{ v^n_j + v^n_{j \pm 1} }{2}. $$ We remarked that for our numerical simulations, it is not necessary to decompose the term $$v \partial_x \zeta$$ (see Remark 3.1) to get the numerical convergence. Indeed, it seems that since the time step is chosen very small, we obtain a solution of the dispersive equation for each iteration. In this case, we do not need to evaluate the term $$\partial_{x} \zeta {\mathcal{H}_{\mu}}^{2} v$$. To ensure the validity of our numerical simulations, we have to be careful of the numerical instability, as to why the time and the space steps are chosen in a way that the following condition is satisfied: \begin{equation} \label{CFL} |v | \frac{{\it{\Delta}} t}{{\it{\Delta}} x} < 1. \end{equation} (5.3) 5.1 Example $$1$$: Convergence curve In this example, we consider the following initial data: \begin{equation*} \zeta_0(x) = {\mathrm{sech}} \left(\frac{\sqrt{3}}{2} x \right), \quad v_0= \zeta_0, \end{equation*} with two different bathymetries: a bump ($$b(x) = \cos(x)$$) and a ripple bottom $$ b(x) = \left\{ \begin{array}{@{}l@{}} 0.5 - \frac{1}{18} (x-8)^2 \mbox{ if } 5 \leq x \leq 11 \\[6pt] 0 \mbox{ otherwise}. \end{array} \right. $$ Note that to avoid numerical reflections due to the boundaries and justify the use of the FFT, we decide to take rapidly decreasing initial data. Figures 1 and 2 display the evolution for different times of the free surface $$\zeta$$ for these two test cases. We decided to take $$\epsilon = 0.1, \mu = 1, \beta = \frac{1}{2}$$, $$N = 2^8, {\it{\Delta}} x = 2L/N, T = 10$$, where N is the mesh modes number, $$L=30$$ the length of the domain and $$T$$ the final time. Note that the time step $${\it{\Delta}} t $$ is chosen iteratively in a way that the CFL condition (5.3) is satisfied. Fig. 1. View largeDownload slide Upper: Evolution of the free surface for different times t = 2.5, 5, 7.5 and 10. Lower: Bottom topography and initial condition. Fig. 1. View largeDownload slide Upper: Evolution of the free surface for different times t = 2.5, 5, 7.5 and 10. Lower: Bottom topography and initial condition. Fig. 2. View largeDownload slide Upper: Evolution of the free surface for different times t = 2.5, 5, 7,5 and 10. Lower: Bottom topography and initial condition. Fig. 2. View largeDownload slide Upper: Evolution of the free surface for different times t = 2.5, 5, 7,5 and 10. Lower: Bottom topography and initial condition. Figures 3 displays the convergence curve for this example. We plot the logarithm of the error (in norm $$H^1 \times L^2$$) in function of the logarithm of the time step $${\it{\Delta}} t.$$ The convergence numerical order is then given by the slope of this curve. For reference, a small line (the dashed line) of slope 1 is added in this figure. We see that the numerical rate of convergence is greater than 1. Fig. 3. View largeDownload slide Convergence curve (for the $$H^1 \times L^2$$-norm) for the Lie method for two bottoms: bump (lower line) and ripple bottom (higher line) for $$T=10.$$ Fig. 3. View largeDownload slide Convergence curve (for the $$H^1 \times L^2$$-norm) for the Lie method for two bottoms: bump (lower line) and ripple bottom (higher line) for $$T=10.$$ 5.2 Example $$2$$: Nonsmooth topographies In this example, we study the evolution of water waves over a rough bottom. This problem is still a mathematical issue. Many models derived from the Euler equations suppose that the bathymetry is smooth. Even worse, a nonsmooth bathymetry introduces singular terms in these models. This issue is particularly easy to see for shallow water models. To handle this, Hamilton (1977) and Nachbin (2003) used a conformal mapping to derive long wave models. Notice also the work of Cathala (2014) who derived alternatives Saint-Venant equations and Boussinesq systems with nonsmooth topographies that do not involve any singular terms. We notice that the Saut–Xu equations (1.1) can handle a nonsmooth topography (see Theorem (2.2)) and our numerical scheme too (see Theorem (4.6)). In the following, we give an example with a nonsmooth bathymetry. We consider the following initial conditions and bathymetry \begin{equation*} \zeta_{0}(x) = v_{0}(x) = e^{-x^{2}} \text{ and } b(x)=\frac{\beta}{4}(1+\tanh(100(x-2)))(1-\tanh(100(x-8))). \end{equation*} We decided to take $$\mu =1$$, $$\beta=0.5$$, $$\epsilon=0.1$$, $$N = 2^8, {\it{\Delta}} x = 2L/N$$, where N is the mesh modes number and $$L=20$$ the length of the domain. The time step $${\it{\Delta}} t $$ is chosen iteratively in a way that the CFL condition (5.3) is satisfied. Figure 4 displays the evolution of the surface $$\zeta$$ for different times over the bottom. Fig. 4. View largeDownload slide Evolution of the free surface (higher lines) for different times t = 0, 3, 6, 9 and 12 over a rough bottom (lower line). Fig. 4. View largeDownload slide Evolution of the free surface (higher lines) for different times t = 0, 3, 6, 9 and 12 over a rough bottom (lower line). 5.3 Example $$3$$: Boussinesq regime In Section 3, we crucially use the fact that $$\mu$$ is bounded from below. In this example, we test our scheme for small values of $$\mu$$ (also called the shallow water regime). We show that our scheme is still valid even if we do not have a proof of the convergence of our scheme in this regime. In the shallow water regime, there is a huge literature for asymptotic models (see, e.g., Lannes, 2013). Among all these asymptotic models, we have the KdV equation. It is a model obtained under the Boussinesq regime, i.e., when $$\epsilon = \mu$$, $$\beta = 0$$ and $$\mu$$ small. In the following, we formally derive a KdV equation from the Saut–Xu equations, and we give numerical simulations in this setting. We recall that without the assumption $$\nu = \frac{1}{\mu}$$, the Saut–Xu equations are given by the system (2.3). Notice also that \begin{equation}\label{dt_Hmu} \mathcal{H}_{\mu} = - \sqrt{\mu} \partial_{x} - \frac{1}{3} \mu^{\frac{3}{2}} \partial_{x}^{3} + \mathcal{O}(\mu^{2}). \end{equation} (5.4) Then, if we assume that $$\mu = \epsilon$$, $$\nu=1$$ (since $$\nu \sim 1$$ if $$\mu$$ is small) and we drop all the terms of order $$\mathcal{O}(\mu^{2})$$ in System (2.3), we obtain the following equations \begin{equation}\label{saut-xu_boussi} \left\{ \begin{aligned} &\partial_{t} \zeta + \partial_{x} v + \mu v \partial_{x} \zeta - \frac{1}{2} \mu^{\frac{3}{2}} v \partial_{x} \zeta + \frac{1}{3} \mu \partial_{x}^{3} v + \mu \zeta \partial_{x} v =0,\\ &\partial_{t} v + \partial_{x} \zeta + \mu v \partial_{x} v + \mu^{\frac{3}{2}} \frac{1}{2} v \partial_{x} v= 0. \end{aligned} \right. \end{equation} (5.5) Formally, the solutions of this system are close to the solutions of (2.3) with an accuracy of order $$\mathcal{O}(\mu^{2})$$. Notice that this system is not a standard Boussinesq system (in the sense of Bona et al. (2002) or Lannes (2013)) because of our nonlinear change of variables (2.2). Using the approach developed in Schneider & Wayne (2012), Bona et al. (2005) and Alvarez-Samaniego & Lannes (2008) (see also Lannes, 2013, Part 7.1.1), we can check that, formally, the following KdV equation is an asymptotic model of the system (5.5) \begin{equation}\label{kdv_eq} \partial_{t} f + \frac{3}{2} f \partial_{x} f + \frac{1}{6} \partial_{x}^{3} f = 0. \end{equation} (5.6) This means that if we solve (5.5) with the initial data $$\left(f_{0}, f_{0} \right)$$ and (5.6) with the initial datum $$f_{0}$$, the solution $$\left(\zeta, v \right)(t,x)$$ of (5.5) is close to $$\left(f,f \right) (\mu t,x-t)$$. Furthermore, if we take $$f_{0}(x) = \alpha \text{sech}^{2} \left(\sqrt{\frac{3}{4} \alpha} x \right)$$, the solution $$f$$ of the KdV equation with this initial datum is the soliton $$f(t,x) = f_{0}(x-ct)$$ with $$c=\frac{\alpha}{2}$$. Hence, in this case, the solutions of (5.5) and (2.3) are close to a soliton. In the following, we check that the solution to (1.1) is indeed close to the KdV solution when $$\mu$$ is small. We simulate one soliton. We took $$v_0(x) = \zeta_0(x) = {\mathrm{sech}}^2 \left(\frac{\sqrt{3}}{2} x \right)$$, $$\epsilon = \mu = 0.01$$, $$\alpha = 1$$ and the final time is $$T=10$$. We decided to take $$N = 2^9, {\it{\Delta}} x = 2L/N$$, where N is the mesh modes number and $$L=30$$ the length of the domain. The time step $${\it{\Delta}} t $$ is chosen iteratively in a way that the CFL condition (5.3) is satisfied. Figure 5 represents the evolution of this soliton at different times. Hence, our scheme is still valid when $$\mu$$ is small. Fig. 5. View largeDownload slide Evolution of the soliton at different times $$t=0, 3, 6, 9$$ ($$\epsilon = 0.01$$). Fig. 5. View largeDownload slide Evolution of the soliton at different times $$t=0, 3, 6, 9$$ ($$\epsilon = 0.01$$). In deep water ($$\mu$$ not small), the KdV approximation ceases to be a good approximation. To get some insight on the range of validity of the KdV approximation, we compare in Fig. 6 the solution of (1.1) with the exact soliton after a time $$T=10$$ for various values of $$\mu$$. We took the same numerical parameters that were taken before. We notice that even for $$\epsilon = \mu=0.1$$ and a final time $$T=\frac{1}{\mu}$$, the KdV approximation remains a good approximation of the Saut–Xu system. Fig. 6. View largeDownload slide Difference after a time $$T=10$$ between a real soliton and a soliton generated by our scheme with the same initial data and for different values of $$\epsilon=\mu$$. Abscissa: value of $$\epsilon$$; Ordinate: quotient of the difference after a final time $$T=10$$ by the maximum of the soliton. Fig. 6. View largeDownload slide Difference after a time $$T=10$$ between a real soliton and a soliton generated by our scheme with the same initial data and for different values of $$\epsilon=\mu$$. Abscissa: value of $$\epsilon$$; Ordinate: quotient of the difference after a final time $$T=10$$ by the maximum of the soliton. 5.4 Example $$4$$: Rapidly varying topographies In this example, we study the evolution of water waves over a rapidly varying periodic bottom. We assume that $$\mu =1$$. This problem is linked to the Bragg reflection phenomenon (see, e.g., Mei, 1985; Liu & Yue, 1998; Guyenne & Nicholls, 2007–08). We take \begin{equation}\label{ini_data_rapidly_bot} \zeta_{0} = v_{0} = \text{sech}^{2}\left(\frac{\sqrt{3}}{2} x \right) \text{ and } b(x)=\cos(\alpha x). \end{equation} (5.7) We decided to take $$N = 2^9, {\it{\Delta}} x = 2L/N$$, where N is the mesh mode number and $$L=30$$ the length of the domain. The time step $${\it{\Delta}} t $$ is chosen iteratively in a way that the CFL condition (5.3) is satisfied. Figure 7 displays the difference between the case of a flat bottom and the case of a bottom of the form $$b(x)=\cos(\alpha x)$$ for different values of $$\alpha$$. Figure 8 compares the evolution of water waves when we take $$\alpha=10$$ (blue line) and when we take $$b(x)=0$$ (red line). We observe a homogenization effect when $$\alpha$$ is large. It seems that when $$\alpha$$ goes to infinity, a solution of the Saut–Xu equations converges to a solution of the Saut–Xu equations with a flat bottom (corresponding to the mean of $$b$$). Notice that this result is different from what we could see in the literature (e.g., Chupin, 2012; Craig et al., 2012), since we take a bottom of the form $$b(x)=\cos (\alpha x)$$ and not of the form $$b(x)= \frac{1}{\alpha} \cos(\alpha x)$$. Our numerical simulations suggest therefore a homogenization effect for large amplitude bottom variations that has not been investigated so far. Fig. 7. View largeDownload slide Difference between a water wave over a rapidly varying topography $$b(x)=\cos(\alpha x)$$ and a water wave over a flat bottom. Abscissa: value of $$\alpha$$; Ordinate: quotient of the difference after a final time $$T=10$$ by the maximum of $$\zeta_{0}$$. Fig. 7. View largeDownload slide Difference between a water wave over a rapidly varying topography $$b(x)=\cos(\alpha x)$$ and a water wave over a flat bottom. Abscissa: value of $$\alpha$$; Ordinate: quotient of the difference after a final time $$T=10$$ by the maximum of $$\zeta_{0}$$. Fig. 8. View largeDownload slide Comparison between the evolution of a water wave over a bottom of the form $$b(x)=\cos(10x)$$ (lower line) and the evolution of a water wave over a flat bottom after a time T = 10. $$\epsilon =0.05$$, $$\beta=0.5$$. The two different surfaces are very close. Fig. 8. View largeDownload slide Comparison between the evolution of a water wave over a bottom of the form $$b(x)=\cos(10x)$$ (lower line) and the evolution of a water wave over a flat bottom after a time T = 10. $$\epsilon =0.05$$, $$\beta=0.5$$. The two different surfaces are very close. Funding ANR project Dyficolti [ANR-13-BS01-0003 to B.M.]. References Alinhac, S. & Gérard, P. ( 1991 ) Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs Actuels. [Current Scholarship]. InterEditions. Paris, France : Éditions du Centre National de la Recherche Scientifique (CNRS), Meudon . Alvarez-Samaniego, B. & Lannes, D. ( 2008 ) Large time existence for 3D water-waves and asymptotics. Invent. 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Google Scholar CrossRef Search ADS Appendix In this part, we give some estimate for the operator $${\mathcal{H}_{\mu}}$$ and some standard product and commutator estimates. For the estimates for $${\mathcal{H}_{\mu}}$$, we refer to Saut & Xu (2012, Part III). For the other estimates, we refer to Alinhac & Gérard (1991) and Lannes (2006). We recall that $${\mathcal{H}_{\mu}}$$ is defined by \begin{equation*} {\mathcal{H}_{\mu}} = - \frac{\tanh(\sqrt{\mu} D)}{D} \partial_{x}. \end{equation*} First, we show that $${\mathcal{H}_{\mu}}$$ is a zero-order operator. Proposition A.1 Let $$s \geq 0$$ and $$\mu$$ satisfying Condition (1.3). Then, \begin{equation*} \left\lvert {\mathcal{H}_{\mu}} u \right\rvert_{H^{s}} \leq C \left(\mu_{\max} \right) \left\lvert u \right\rvert_{H^{s}}\!. \end{equation*} Furthermore, for all $$s \geq r \geq 0$$, \begin{equation*} \left\lvert \left( {\mathcal{H}_{\mu}}^{2} + 1 \right) u \right\rvert_{H^{s}} \leq C_{r} \left(\frac{1}{\mu_{\min}} \right) \left\lvert u \right\rvert_{H^{r}}\!. \end{equation*} Then, we give a commutator estimate for $${\mathcal{H}_{\mu}}$$. Proposition A.2 Let $$s \geq 0$$, $$t_{0} > \frac{1}{2}$$, $$r \geq 0$$ and $$\mu$$ satisfying Condition (1.3). Then, \begin{equation*} \left\lvert \left[{\mathcal{H}_{\mu}}, a \right] u \right\rvert_{2} \leq C \left\lvert a \right\rvert_{H^{t_{0}}} \left\lvert f \right\rvert_{2}, \end{equation*} \begin{equation*} \left\lvert |\xi|^{s} \widehat{\left[{\mathcal{H}_{\mu}}, a \right] u} \right\rvert_{2} \leq C \left(\frac{1}{\mu_{\min}} \right) \left\lvert a \right\rvert_{H^{r+s}} \left\lvert \frac{\left(1+|\xi| \right)^{t_{0}}}{|\xi|^{r}} \widehat{u} \right\rvert_{2} \end{equation*} and \begin{equation*} \left\lvert |\xi|^{s} \widehat{\left[{\mathcal{H}_{\mu}}, a \right] u} \right\rvert_{2} \leq C \left(\frac{1}{\mu_{\min}} \right) \left\lvert a \right\rvert_{H^{r+s+t_{0}}} \left\lvert \frac{1}{|\xi|^{r}} \widehat{u} \right\rvert_{2}. \end{equation*} We recall the well-known Coifman–Meyer estimate. We recall also that $${\it{\Lambda}}$$ is the Fourier multiplier $${\it{\Lambda}} = \sqrt{1+D^{2}}$$. Proposition A.3 Let $$s > \frac{3}{2}$$, $$u \in H^{s}({\mathbb{R}})$$ and $$v \in H^{s-1}({\mathbb{R}})$$. Then, we have the following commutator estimate \begin{equation*} \left\lvert \left[ {\it{\Lambda}}^{s}, u \right] v \right\rvert_{2} \leq C \left\lvert u \right\rvert_{H^{s}} \left\lvert v \right\rvert_{H^{s-1}}\!. \end{equation*} We recall also the following product estimate. Proposition A.4 Let $$s_{1}, s_{2},s$$ such that $$s_{1} + s_{2} \geq 0$$, $$s \leq \min \left( s_{1}, s_{2} \right)$$ and $$s < s_{1} + s_{2} - \frac{1}{2}$$. Let $$u \in H^{s_{1}}({\mathbb{R}})$$ and $$v \in H^{s_{2}}({\mathbb{R}})$$. Then, \begin{equation*} \left\lvert u v \right\rvert_{H^{s}} \leq C \left\lvert u \right\rvert_{H^{s_{1}}} \left\lvert v \right\rvert_{H^{s_{2}}}\!. \end{equation*} © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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IMA Journal of Numerical AnalysisOxford University Press

Published: Jul 29, 2017

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