paper

High Order Lagrangian ADER-WENO Schemes on Unstructured Meshes - Application of Several Node Solvers to Hydrodynamics and Magnetohydrodynamics

arXiv:1310.7256 · doi:10.1002/fld.3947

Abstract

In this paper we present a class of high order accurate cell-centered Arbitrary-Eulerian-Lagrangian (ALE) one-step ADER-WENO finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space-time Galerkin predictor allows the schemes to be high order accurate also in time by using an element-local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained i) either as an arithmetic average among the states surrounding the node, or, ii) as a solution of multiple one-dimensional half-Riemann problems around a vertex, or, iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional HLL Riemann. Once the vertex velocity and thus the new node location has been determined by the node solver, the local mesh motion is then constructed by straight edges connecting the vertex positions at the old time level with the new ones at the next time level. If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. We apply the high order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic MHD equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration.

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