Bulletin of the Brazilian Mathematical Society, New Series

Abreu, Eduardo; Bustos, Abel; Lambert, Wanderson

doi: 10.1007/s00574-016-0118-1pmid: N/A

We developed an unsplitting finite volume scheme to account the delicate nonlinear balance between numerical approximations of the hyperbolic flux function and the source linked to balance laws. The method is Riemann-solver-free and no upwinding technique is used. By means of this new approach, we conducted an analysis for two new models of balance laws linked to compositional and thermal flow in porous media problems, under and without a thermodynamic equilibrium hypothesis. For concreteness, we adopt the nitrogen and steam injection models in a porous media. To this model we found an interesting behavior linked to the relaxation term, which is the existence of a non-monotonic traveling wave. We applied this numerical technique to others well-known differential models with relaxation terms available in the literature. Qualitatively we were able to reproduce the expected results.

Adimurthi, ; Kumar Kenettinkara, Sudarshan; Veerappa Gowda, G.

doi: 10.1007/s00574-016-0119-0pmid: N/A

A second order scheme is constructed for the scalar conservation laws with flux function allowed to be discontinuous in the space variable. The corresponding numerical solutionis shown to converge to the (A,B) entropysolution. Numerical results are presented to validate the convergence analysis. In addition, it is inferred that these numerical solutions are comparable with those obtained using the standard minmod limiter.

Aggarwal, Aekta; Goatin, Paola

doi: 10.1007/s00574-016-0120-7pmid: N/A

We present a Lax-Friedrichs type scheme to compute the solutions of a class of non-local and non-linear systems of conservation laws in several space dimensions. The convergence of the approximate solutions is proved by providing suitable L 1 , L ∞ and BV uniform bounds. To illustrate the performances of the scheme, we consider an application to crowd dynamics. Numerical integrations show the formation of lanes in groups moving in opposite directions. This is joint work with R.M. Colombo (INDAM Unit, University of Brescia).

Alla, Alessandro; Falcone, Maurizio; Kalise, Dante

doi: 10.1007/s00574-016-0121-6pmid: N/A

We propose a computational approach for the solution of an optimal control problem governed by the wave equation. We aim at obtaining approximate feedback laws by means of the application of the dynamic programming principle. Since this methodology is only applicable for low-dimensional dynamical systems, we first introduce a reduced-order model for the wave equation by means of Proper Orthogonal Decomposition. The coupling between the reduced-order model and the related dynamic programming equation allows to obtain the desired approximation of the feedback law. We discuss numerical aspects of the feedback synthesis and providenumerical tests illustrating this approach.

Amadori, Debora; Baiti, Paolo; Corli, Andrea; Dal Santo, Edda

doi: 10.1007/s00574-016-0122-5pmid: N/A

The paper deals with a simple nonlinear hyperbolic system of conservation laws modeling the flow of an inviscid fluid. The model is given by a standard isothermal p-system of the gasdynamics, for which phase transitions of the fluid are taken into consideration via a third homogeneous equation. We focus on the case of initial data consisting of two different phases separated by an interface. By means of an adapted version of the front tracking algorithm, we prove the global-in time existence of weak entropic solutions under suitable assumptions on the (possibly large) initial data.

Andrade, P.; de Souza, A.; Furtado, F.; Marchesin, D.

doi: 10.1007/s00574-016-0123-4pmid: N/A

In this work we present the construction of the Riemann solution for a system of two conservation laws representing displacement in immiscible three-phase flow. The porousmedium is initially filled with oil and small amounts of water and gas; then a fixed proportion of water and gas is injected. We use the wave curve method to determine the wave sequences in the Riemann solution for arbitrary initial and injection data in the above mentioned class. We show the L Loc 1 -stability of the Riemann solution with variation of data. We do not verify uniqueness of the Riemann solution, but we believe that it is valid.

Bernstein, Andrew; Chertock, Alina; Kurganov, Alexander

doi: 10.1007/s00574-016-0124-3pmid: N/A

Finite-volume central-upwind schemes for shallow water equations were proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), 133–160]. These schemes are capable of maintaining “lake-at-rest” steady states and preserving the positivity of the computed water depth. The well-balanced and positivity preserving features of the central-upwind schemes are achieved, in particular, by using continuous piecewise linear interpolation of the bottom topography function. However, when the bottom function is discontinuous or a model with a moving bottom topography is studied, the continuous piecewise linear approximationmay not be sufficiently accurate and robust.

Berres, Stefan; Castañeda, Pablo

doi: 10.1007/s00574-016-0125-2pmid: N/A

This contribution is a condensed version of an extended paper, where a contact manifold emerging in the interior of the phase space of a specific hyperbolic system of two nonlinear conservation laws is examined. The governing equations are modelling bidisperse suspensions, which consist of two types of small particles differing in size and viscosity that are dispersed in a viscous fluid. Based on the calculation of characteristic speeds, the elementary waves with the origin as left Riemann datum and a general right state in the phase space are classified. In particular, the dependence of the solution structure of this Riemann problem on the contact manifold is elaborated.

Berthon, C.; Chalons, C.; Cornet, S.; Sperone, G.

doi: 10.1007/s00574-016-0126-1pmid: N/A

The present work is focused on the numerical approximation of the shallow water equations. When studying this problem, one faces at least two important issues, namely the ability of the scheme to preserve the positiveness of the water depth, along with the ability to capture the stationary states.We propose here aGodunov-typemethod that fully satisfies the previous conditions, meaning that the method is in particular able to preserve the steady states with non-zero velocity.

Bianchini, Stefano; Spinolo, Laura

doi: 10.1007/s00574-016-0127-0pmid: N/A

This note aims at providing an overview of some recent results concerning the viscous approximation of so-called boundary Riemann problems for nonlinear systems of conservation laws in small total variation regimes.