Low Mach Asymptotic Preserving Scheme for the Euler-Korteweg Model

TL;DR

This paper introduces an all-speed, asymptotic preserving finite difference scheme for the Euler-Korteweg model, effectively handling low Mach number regimes without timestep restrictions.

Contribution

It develops a semi-implicit discretization that remains stable and accurate as the Mach number approaches zero, ensuring convergence to the incompressible limit.

Findings

Scheme is asymptotic preserving for low Mach numbers

Stable and efficient for all speed regimes

Converges to incompressible limit as Mach number tends to zero

Abstract

We present an all speed scheme for the Euler-Korteweg model. We study a semi-implicit time-discretisation which treats the terms, which are stiff for low Mach numbers, implicitly and thereby avoids a dependence of the timestep restriction on the Mach number. Based on this we present a fully discrete finite difference scheme. In particular, the scheme is asymptotic preserving, i.e., it converges to a stable discretisation of the incompressible limit of the Euler-Korteweg model when the Mach number tends to zero.