Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld High Order Lagrangian ADER-WENO Schemes on Unstructured Meshes – Application of Several Node Solvers to Hydrodynamics

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 along the principles laid out in [1]. 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, as suggested by Cheng and Shu [2], or, ii) as a solution of multiple one–dimensional half-Riemann problems around a vertex, as suggested by Maire [3], or, iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional HLL Riemann solver recently proposed by Balsara et al. [4]. 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 tn with the new ones at the next time level tn+1. If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unneccesary, since the ALE fluxes already properly take into account th