Asymptotic-Preserving and Well-Balanced Linearly Implicit IMEX Schemes for the Anelastic Limit of the Isentropic Euler Equations with Gravity

TL;DR

This paper develops high-order linearly implicit IMEX schemes that are asymptotic preserving and well-balanced for the anelastic limit of the isentropic Euler equations with gravity, ensuring accuracy in low Mach regimes.

Contribution

It introduces a novel combination of penalization, balance-preserving reconstruction, and source discretization for efficient, accurate simulations of low Mach number flows with gravity.

Findings

Schemes are asymptotic preserving in the zero-Mach limit.

Well-balanced property ensures accuracy near steady states.

Numerical results confirm theoretical properties.

Abstract

We consider the compressible Euler system with anelastic scaling, modeling isentropic flows under the influence of gravity. In the zero-Mach-number limit, the solution of the compressible Euler system converges to a variable density anelastic incompressible limit system. In this work, we present the design and analysis of a class of higher-order linearly implicit IMEX Runge-Kutta schemes that are asymptotic preserving, i.e., they respect the transitory nature of the governing equations in the limit. The presence of gravitational potential warrants the incorporation of the well-balancing property. The scheme is developed as a novel combination of a penalization of a linear steady state, a finite-volume balance-preserving reconstruction, and a source term discretization preserving steady states. The penalization plays a crucial role in obtaining a linearly implicit scheme, and well-balanced flux-source discretization ensures accuracy in very low Mach number regimes. Some results of numerical case studies are presented to corroborate the theoretical assertions.