Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers

摘要

In this paper we use the genuinely multidimensional HLL Riemann solversrecently developed by Balsara et al. to construct a new class ofcomputationally efficient high order Lagrangian ADER-WENO one-step ALE finitevolume schemes on unstructured triangular meshes. A nonlinear WENOreconstruction operator allows the algorithm to achieve high order of accuracyin space, while high order of accuracy in time is obtained by the use of anADER time-stepping technique based on a local space-time Galerkin predictor.The multidimensional HLL and HLLC Riemann solvers operate at each vertex of thegrid, considering the entire Voronoi neighborhood of each node and allows forlarger time steps than conventional one-dimensional Riemann solvers. Theresults produced by the multidimensional Riemann solver are then used twice inour one-step ALE algorithm: first, as a node solver that assigns a uniquevelocity vector to each vertex, in order to preserve the continuity of thecomputational mesh; second, as a building block for genuinely multidimensionalnumerical flux evaluation that allows the scheme to run with larger time stepscompared to conventional finite volume schemes that use classicalone-dimensional Riemann solvers in normal direction. A rezoning step may benecessary in order to overcome element overlapping or crossing-over. We applythe method presented in this article to two systems of hyperbolic conservationlaws, namely the Euler equations of compressible gas dynamics and the equationsof ideal classical magneto-hydrodynamics (MHD). Convergence studies up tofourth order of accuracy in space and time have been carried out. Severalnumerical test problems have been solved to validate the new approach.