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

摘要

In this paper we present a class of high order accurate cell-centeredArbitrary-Eulerian-Lagrangian (ALE) one-step ADER-WENO finite volume schemesfor the solution of nonlinear hyperbolic conservation laws on two-dimensionalunstructured triangular meshes. High order of accuracy in space is achieved bya WENO reconstruction algorithm, while a local space-time Galerkin predictorallows the schemes to be high order accurate also in time by using anelement-local weak formulation of the governing PDE on moving meshes. The meshmotion can be computed by choosing among three different node solvers, whichare for the first time compared with each other in this article: the nodevelocity may be obtained i) either as an arithmetic average among the statessurrounding the node, or, ii) as a solution of multiple one-dimensionalhalf-Riemann problems around a vertex, or, iii) by solving approximately amultidimensional Riemann problem around each vertex of the mesh using thegenuinely multidimensional HLL Riemann. Once the vertex velocity and thus thenew node location has been determined by the node solver, the local mesh motionis then constructed by straight edges connecting the vertex positions at theold time level with the new ones at the next time level. If necessary, arezoning step can be introduced here to overcome mesh tangling or highlydeformed elements. We apply the high order algorithm presented in this paper tothe Euler equations of compressible gas dynamics as well as to the idealclassical and relativistic MHD equations. We show numerical convergence resultsup to fifth order of accuracy in space and time together with some classicalnumerical test problems for each hyperbolic system under consideration.