Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Tetrahedral Meshes for Conservative and Nonconservative Hyperbolic Systems in 3D

摘要

In this paper we present a new family of high order accurateArbitrary-Lagrangian-Eulerian (ALE) one-step ADER-WENO finite volume schemesfor the solution of nonlinear systems of conservative and non-conservativehyperbolic partial differential equations with stiff source terms on movingtetrahedral meshes in three space dimensions. A WENO reconstruction techniqueis used to achieve high order of accuracy in space, while an element-localspace-time Discontinuous Galerkin finite element predictor on moving meshes isused to obtain a high order accurate one-step time discretization. Within thespace-time predictor the physical element is mapped onto a reference elementusing an isoparametric approach, where the space-time basis and test functionsare given by the Lagrange interpolation polynomials passing through apredefined set of space-time nodes. Since our algorithm is cell-centered, thefinal mesh motion is computed by using a suitable node solver algorithm. Arezoning step as well as a flattener strategy are used in some of the testproblems to avoid mesh tangling or excessive element deformations that mayoccur when the computation involves strong shocks or shear waves. We apply ournew high order unstructured ALE schemes to the 3D Euler equations ofcompressible gas dynamics, for which a set of classical numerical test problemshas been solved and for which convergence rates up to sixth order of accuracyin space and time have been obtained. We furthermore consider the equations ofclassical ideal magnetohydrodynamics (MHD) as well as the non-conservativeseven-equation Baer-Nunziato model of compressible multi-phase flows with stiffrelaxation source terms.