High order ADER schemes for continuum mechanics

TL;DR

This paper reviews high order ADER schemes for continuum mechanics, introduces a unified hyperbolic formulation for fluids and solids, and demonstrates their application to moving boundary problems using diffuse interface methods.

Contribution

It presents a modern ADER scheme variant with a posteriori subcell limiting and applies the unified SHTC continuum mechanics model to moving boundary problems.

Findings

ADER schemes achieve high-order accuracy on fixed and moving meshes.

Unified SHTC model effectively describes fluid and solid mechanics.

Diffuse interface approach handles free surface and moving boundary problems.

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.