A space-time expansion discontinuous Galerkin scheme with local time-stepping for the ideal and viscous MHD equations

Abstract

Summary form only given. During the last years the discontinuous Galerkin (DG) schemes became more popular, since they combine the flexibility in handling complex geometries, h/p-adaptivity and efficiency of parallel implementation. Recently, Lorcher et al. developed a DG scheme which allows local time-stepping for unsteady calculations. This scheme is based on a Taylor expansion in space and time (STE) about the barycenter of each cell at the old time level. All time and mixed space-time derivatives are replaced by space derivatives with the help of the so-called Cauchy-Kovalevskaya procedure by making use of the evolution equations. The calculation of the diffusive flux is done through a diffusive generalized Riemann problem (dGRP), introduced by Gassner et al. The local time-stepping strategy allows each cell to have its own time step whereas the high order of accuracy in time is retained. This may significantly speed up calculations. In this talk we present the extension of this scheme to handle the ideal as well as the viscous magnetohydrodynamic equations according to Warburton and Karniadakis. The artificial viscosity limiter developed by Persson and Peraire is used to enable the shock-capturing property. Two-dimensional MHD problems such as the Orszag-Tang vortex or the magnetic blast problem are performed to challenge the capabilities of the proposed space-time expansion scheme.