High Order ADER Schemes for Continuum Mechanics

Abstract

In this paper we first review the development of high order ADER finite\nvolume and ADER discontinuous Galerkin schemes on fixed and moving meshes,\nsince their introduction in 1999 by Toro et al. We show the modern variant of\nADER based on a space-time predictor-corrector formulation in the context of\nADER discontinuous Galerkin schemes with a posteriori subcell finite volume\nlimiter on fixed and moving grids, as well as on space-time adaptive Cartesian\nAMR meshes. We then present and discuss the unified symmetric hyperbolic and\nthermodynamically compatible (SHTC) formulation of continuum mechanics\ndeveloped by Godunov, Peshkov and Romenski (GPR model), which allows to\ndescribe fluid and solid mechanics in one single and unified first order\nhyperbolic system. In order to deal with free surface and moving boundary\nproblems, a simple diffuse interface approach is employed, which is compatible\nwith Eulerian schemes on fixed grids as well as direct\nArbitrary-Lagrangian-Eulerian methods on moving meshes. We show some examples\nof moving boundary problems in fluid and solid mechanics.\n