Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Divergence-free discontinuous Galerkin method for MHD and Maxwell's equations

Prof. Dr. Praveen Chandrashekar talk

**Date:** 10.04.19 **Time:** 16.15 - 17.45 **Room:** Y27H25

Some PDE models like MHD and Maxwell's equations contain magnetic field as a dependent variable which must be divergence-free due to the non-existence of magnetic monopoles. This is an inherent constraint satisfied by the induction equation due to its curl structure. Numerical schemes may not preserve this structure unless they are specifically designed for this purpose. A staggered storage of variables is useful to satisfy such constraints by a numerical scheme. In this talk, I will describe two approaches to construct high order numerical approximations based on discontinuous Galerkin method that are constraint preserving. In the first approach, we perform a divergence-free reconstruction of the magnetic field while in the second approach, the divergence constraint is automatically satisfied by the numerical scheme due to the use of H(div) elements. The numerical flux used in such DG methods must satisfy a consistency condition between the 1-D and multi-D Riemann solvers, and we construct HLL-type schemes for MHD that exhibit such consistency. These methods are useful in applications where explicit time stepping schemes can be used and I will show some results for MHD and Maxwell's equations.