Arbitrary-Lagrangian–Eulerian ADER–WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws

Abstract

In this article a new high order accurate cell-centered Arbitrary-Lagrangian–Eulerian (ALE) Godunov-type finite volume method with time-accurate local time stepping (LTS) is presented. The method is by construction locally and globally conservative. The scheme is based on a one-step predictor–corrector methodology in space–time and uses three main building blocks: First, a high order piecewise polynomial WENO reconstruction, to obtain a high order data representation in space from the known cell averages of the underlying finite volume scheme. Second, a high order space–time Galerkin predictor step based on a weak formulation of the governing PDE on moving control volumes. Third, a high order one-step finite volume scheme, based directly on the integral formulation of the conservation law in space–time. The algorithm being entirely based on space–time control volumes naturally allows for hanging nodes also in time, hence in this framework the implementation of a consistent and conservative time-accurate LTS becomes very natural and simple. The method is validated on some classical shock tube problems for the Euler equations of compressible gas dynamics and the magnetohydrodynamics equations (MHD). The performance of the new scheme is compared with a classical high order ALE finite volume scheme based on global time stepping. To the knowledge of the author, this is the first high order accurate Lagrangian finite volume method ever presented together with a conservative and time-accurate local time stepping feature.