Asymptotic Preserving and Multiscale Methods for Kinetic and Hyperbolic Problems

Asymptotic preserving numerical schemes for singular hyperbolic PDE's

Sebastian Noelle



I will discuss systems of conservation laws where some wave speeds become singular. The classic example is the low Mach number limit in gas dynamics. In the singular limit, hyperbolicity gets lost, and near the limit, explicit time discretizations become either inefficient or unstable, both due to the CFL conditions. The established concept to design efficient and stable algorithms near the singular limit is a time-IMplicit-EXplicit splitting, called IMEX. The recent asymptotic preserving (AP) IMEX schemes follow the singular limit. A key question is the asymptotic stability of these schemes. I discuss two examples: a well-known, but unstable, scheme, and an also well-known, but stable scheme. Then I present a new stability analysis for IMEX schemes, which explains the outcomes of these experiments. I also give an outlook to a new concept of splittings, and a first application.