Z. Angew. Math. Phys. (2018) 69:8
2017 Springer International Publishing AG,
part of Springer Nature
published online December 14, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
A well-balanced scheme for Ten-Moment Gaussian closure equations with source term
Asha Kumari Meena and Harish Kumar
Abstract. In this article, we consider the Ten-Moment equations with source term, which occurs in many applications related
to plasma ﬂows. We present a well-balanced second-order ﬁnite volume scheme. The scheme is well-balanced for general
equation of state, provided we can write the hydrostatic solution as a function of the space variables. This is achieved by
combining hydrostatic reconstruction with contact preserving, consistent numerical ﬂux, and appropriate source discretiza-
tion. Several numerical experiments are presented to demonstrate the well-balanced property and resulting accuracy of the
Mathematics Subject Classiﬁcation. 65M08, 65M12, 35L60.
Keywords. Finite volume methods, Well-balanced scheme, Ten-Moment Gaussian closure equations.
Ten-Moment equations are a ﬂuid ﬂow model derived by taking higher moments of Boltzmann equations to
capture anisotropic eﬀects, which are ignored (due to the assumption of local thermodynamic equilibrium)
when we derive Euler equations of compressible ﬂows (see [1,2]). This results in a system of equations,
which includes the evolution of the symmetric pressure tensor. Due to this, the Ten-Moment model is
suitable for several applications, especially related to the plasma ﬂows, where the anisotropic behaviour
of plasma is important (see [1–9]).
The Ten-Moment system is a system of hyperbolic conservation laws (see ). So the system of
equations can exhibit discontinuous solution, even for smooth initial data. Because of this, we need to
consider weak solutions. Furthermore, to choose the physically correct solution, additional criteria in the
form of entropy condition are imposed.
In this article, we consider the Ten-Moment system with a source term where source depends on a
given potential function W (see ). The applications of this model are related to the plasma physics,
where interest is to study the interaction of matter with a laser. One example of such source is to test
the eﬀect of laser on plasma with electron quiver energy given by function W (see ).
The systems of hyperbolic balance laws are often discretized using ﬁnite volume (or ﬁnite diﬀerence)
methods. The higher-order accuracy is obtained by reconstructing the traces on the cell boundary using
TVD limiters, ENO and WENO reconstruction procedures. Source terms are discretized using consistent
methods like central diﬀerence schemes. For most applications, these methods are adequate and provide
good approximations of weak solutions. However, for several applications, we are interested in hydrostatic
solutions of the model. These standard numerical methods fail to preserve such stationary states on coarser
meshes as they produce signiﬁcant errors. We call a numerical scheme well-balanced if it preserves the
nontrivial stationary solutions. So, in general, the standard schemes are not well-balanced. On the other
hand, if the numerical scheme can preserve steady state, we can improve the accuracy of the solutions,
especially for the states close to the stationary solutions. This has several beneﬁts, in particular when we
compute on the coarser mesh.