Abstract
In this paper we first review the development of high order ADER finite volume and ADER discontinuous Galerkin schemes on fixed and moving meshes, since their introduction in 1999 by Toro et al. We show the modern variant of ADER based on a space-time predictor-corrector formulation in the context of ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter on fixed and moving grids, as well as on space-time adaptive Cartesian AMR meshes. We then present and discuss the unified symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of continuum mechanics developed by Godunov, Peshkov and Romenski (GPR model), which allows to describe fluid and solid mechanics in one single and unified first order hyperbolic system. In order to deal with free surface and moving boundary problems, a simple diffuse interface approach is employed, which is compatible with Eulerian schemes on fixed grids as well as direct Arbitrary-Lagrangian-Eulerian methods on moving meshes. We show some examples of moving boundary problems in fluid and solid mechanics.
Highlights
AND REVIEW OF THE ADER APPROACHThe development of high order numerical schemes for hyperbolic conservation laws has been one of the major challenges of numerical analysis for the last decades
In this paper we first review the development of high order ADER finite volume and ADER discontinuous Galerkin schemes on fixed and moving meshes, since their introduction in 1999 by Toro et al We show the modern variant of ADER based on a space-time predictor-corrector formulation in the context of ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter on fixed and moving grids, as well as on space-time adaptive Cartesian adaptive mesh refinement on Cartesian grids (AMR) meshes
We compare the results obtained with a third order ADER-WENO finite volume scheme on moving unstructured Voronoi meshes with possible topology changes, Gaburro et al [77], with those obtained with a fourth order ADER discontinuous Galerkin finite element scheme on a very fine uniform Cartesian mesh composed of 512 × 512 elements, which will be taken as the reference solution for this benchmark
Summary
The development of high order numerical schemes for hyperbolic conservation laws has been one of the major challenges of numerical analysis for the last decades. Three dimensional Euler equations on unstructured meshes, Boscheri and Dumbser [62, 63], including the discretization of non-conservative products Further works in this area involve the use of local timestepping techniques, [64, 65]; coupling with multidimensional HLL Riemann solvers, Boscheri et al [66]; solution of magnetohydrodynamics problems (MHD), [67, 68]; development of a quadrature-free approach to increase the computational efficiency of the overall method, Boscheri and Dumbser [69]; use of curvilinear unstructured meshes, Boscheri and Dumbser [70]; or extension to solve the GPR model, Boscheri et al [71] and Peshkov et al [72].
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