I³MS - Lukacova Seminar
Prof. Dr. Maria Lukacova - Asymptotic Preserving IMEX Schemes for Singulary Perturbed Flows
Department of Mathematics, Johannes Gutenberg University Mainz
In this contribution we present our recent results on the second order asymptotic preserving well-balanced schemes for the Euler equations with the gravity force. In the case of low Mach number flows, which typically arise in geophysical applications, the flow has multiscale behaviour. Our aim is to present and analyse new well-balanced asymptotic preserving scheme that efficiently works for different Mach numbers, in particular also for weakly compressible regime. In order to efficiently resolve slow dynamics we split the whole nonlinear system in a stiff linear part governing the acoustic and gravity waves and a non-stiff nonlinear part that models nonlinear advection effects. For time discretization we use a special class of the so-called globally stiffly accurate IMEX schemes and approximate the stiff linear operator implicitly and the non-stiff nonlinear operator explicitly. Finite volume approximation with the central fluxes and the Rusanov or Lax-Friedrichs numerical fluxes is used for the spatial approximation of the linear and nonlinear subsystem, respectively. We prove theoretically that the methods are asymptotically consistent and asymptotic stable uniformly with respect to small Mach numbers and analyse their convergence rates in the singular limit when the model parameters tend to zero. This work has been obtained in the cooperation with G. Bispen, L. Yelash and S. Noelle, it has been supported by the German Grant Foundation (DFG) under LU 1470/2-3.