Klaus Kaiser

A New Stable Splitting for the Isentropic Euler Equations

In this work, we propose a new way of splitting the flux function of the isentropic compressible Euler equations at low Mach number into stiff and non-stiff parts. Following the IMEX methodology, the latter ones are treated explicitly, while the first ones are treated implicitly. The splitting is based on the incompressible limit solution, which we call reference solution. An analysis concerning the asymptotic consistency and numerical results demonstrate the advantages of this splitting.

The RS-IMEX splitting for the isentropic Euler equations

Approximating solutions to singularly perturbed differential equations necessitates the use of stable integrators. One famous approach is to split the equation into stiff and non-stiff parts. Treating stiff parts implicitly, non-stiff ones explicitly leads to so-called IMEX schemes. These schemes are supposed to exhibit very good accuracy and uniform stability, however, not every (seemingly reasonable) splitting induces a stable algorithm. In this paper, we present a new IMEX-splitting based on a reference solution (RS) applied to the isentropic Euler equations.

Asymptotic error analysis of an IMEX Runge–Kutta method

Abstract We consider a system of singularly perturbed differential equations with singular parameter e ≪ 1 , discretized with an IMEX Runge–Kutta method. The splitting needed for the IMEX method stems from a linearization of the fluxes around the limit solution. We analyze the asymptotic convergence order as e → 0 . We show that in this setting, the stage order of the implicit part of the scheme is of great importance, thereby explaining earlier numerical results showing a close correlation of errors of the splitting scheme and the fully implicit one.

The Influence of the Asymptotic Regime on the RS-IMEX

In this work, we investigate the performance and explore the limits of a novel implicit-explicit splitting (Kaiser and Schutz, A high-order method for weakly compressible flows. Commun. Comput. Phys. 22(4): 1150–1174, 2017) for the efficient treatment of singularly perturbed ODEs. We consider a singularly perturbed ODE where, based on the choice of initial conditions, the unperturbed equation does not necessarily describe the behavior of the perturbed one accurately. For the splitting presented in Kaiser and Schutz, (A high-order method for weakly compressible flows. Commun. Comput. Phys. 22(4): 1150–1174, 2017), this has a tremendous influence as it explicitly depends on the solution to the unperturbed equation. That this indeed poses a problem is shown numerically; but also the remedy of using the ‘correct’ asymptotics is presented. Comparisons with a fully implicit and a standard implicit-explicit splitting are shown.

Asymptotic Preserving Discontinuous Galerkin Method

In this work a numerical method for the low Mach isentropic Navier-Stokes equation is devised. Supported by a short analysis, we observe that this algorithm is capable of treating the low Mach equations also in the limit of zero Mach number. The performance of the algorithm is investigated numerically, where one can observe problems with the order of convergence. We conclude with possible reasons and remedies.