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Application of Space-Time CE/SE Method to Shallow Water Magnetohydrodynamics Equations

by Qamar, S., Warnecke, G..

Series: 2005-17, Preprints

76W05 Magnetohydrodynamics and electrohydrodynamics
76W05 Magnetohydrodynamics and electrohydrodynamics
35L65 Conservation laws
65M06 Finite difference methods
35L45 Initial value problems for first-order hyperbolic systems
35L67 Shocks and singularities

In this article we apply a Space-Time Conservation
Element and Solution Element (CE/SE) method for the approximate solution
of shallow water magnetohydrodynamics (SMHD) equations in
one and two space dimensions. These equations model the dynamics of nearly incompressible
conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium
in the third direction. In this article we are using a variant of the CE/SE method developed by Zhang, Yu and Chang (JCP-175, 2002). This method uses structured and unstructured quadrilateral and hexahedral meshes in two and three space dimensions, respectively.
In this method, a single conservation element at each grid point is employed for solving conservation laws no matter in one, two, and three space dimensions.
The present scheme use the conservation element to calculate flow variables only, while the gradients of flow variables
are calculated by central differencing reconstruction procedure. We give both one- and two-dimensional test computations. A qualitative
comparison reveals an excellent agreement with previous published results of wave propagation method and evolution Galerkin schemes. The
one- and two-dimensional computations reported in this paper demonstrate the remarkable versatility of the
present CE/SE scheme.

Sallow water magnetohydrodynamic equations, CE/SE method, conservation laws, hyperbolic systems, discontinuous solutions.

This paper was published in:
This article will appear in the Journal of Computational and Applied Mathematics (JCAM).