Abstract

The density and pressure are positive physical quantities in magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP) numerical schemes for compressible MHD is highly desired but remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the unclear relations between the PP property and the divergence-free condition on the magnetic field. In this talk, I am going to introduce our recent efforts on understanding, designing and rigorously analyzing PP methods for ideal MHD and relativistic MHD systems. Particularly, important relations between the PP property and divergence-free magnetic field will be revealed from several different viewpoints.