Seminar: Asymptotic Preserving IMEX Runge-Kutta Schemes for Singular Hyperbolic Systems of Conservation Laws

Abstract

Sound waves are considered unimportant for most atmospheric flow applications, yet they provide technical difficulties in numerical solutions of the compressible flow equations in atmospherically relevant flow regimes due to their fast characteristic time scales. In the first half of this talk we present a systematic derivation of some distinguished sound-proof models of atmospheric flows, e.g. the low Mach number model, the Boussinesq model, and the anelastic and quasistatic models, using an asymptotic analysis of the Euler equations of compressible fluids. These sound-proof models have a mixed hyperbolic-elliptic nature in contrast with the purely hyperbolic compressible Euler equations, i.e. these are singular limits.

It is well known from the literature that in the case of low Mach number limit of the compressible Euler equations, compressible flow solvers suffer from several pathologies in a low Mach number
regime – namely, reduction of order, lack of stability and inconsistency. In the second half of the present talk, we consider a framework for analysing the above three difficulties by taking the wave equation system with advection as a prototype model of the Euler equations. Guided by a systematic multi-scale asymptotic expansion, we split the fluxes into a so-called stiff part and a non-stiff part. The non-stiff part is so designed that its Jacobian matrix always has finite eigenvalues, facilitating the use of standard upwind discretisation procedures. On the other hand, the stiff part constitutes a wave equation system with very large wave speeds, amenable for implicit treatments. We propose a second order implicit-explicit (IMEX) Runge-Kutta (RK) scheme for time discretisation, therein the non-stiff part is treated explicitly and the stiff part implicitly. In order to simplify the implicit nature of the scheme and the solution of the resulting algebraic equations, only diagonally implicit RK schemes are studied. We characterise the accuracy of the scheme using its ability to leave a so called well prepared space of constant densities and divergence free velocities invariant. We present the accuracy analysis, L2-stability analysis of the new scheme and the results of few numerical experiments to substantiate the proposed findings.

Date and Time : 25-08-2017 12:55

Venue : CSB 3110

Speaker : Saurav Samantaray, IISER TVM

Leave a Reply

Your email address will not be published. Required fields are marked *