High order asymptotic preserving well-balanced finite difference WENO schemes for all Mach full Euler equations with gravity

TL;DR

This paper introduces a high-order semi-implicit finite difference WENO scheme for the full Euler equations with gravity, ensuring asymptotic preserving and well-balanced properties across all Mach regimes.

Contribution

It develops a novel high-order semi-implicit WENO scheme that is both asymptotic preserving and well-balanced for all Mach numbers in Euler equations with gravity.

Findings

Scheme is asymptotic preserving in the incompressible limit

Scheme accurately captures gravity and acoustic waves

Numerical experiments validate theoretical properties

Abstract

In this paper, we propose a high order semi-implicit well-balanced finite difference scheme for all Mach Euler equations with a gravitational source term. To obtain the asymptotic preserving property, we start from the conservative form of full compressible Euler equations and add the evolution equation of the perturbation of potential temperature. The resulting system is then split into a (non-stiff) nonlinear low dynamic material wave to be treated explicitly, and (stiff) fast acoustic and gravity waves to be treated implicitly. With the aid of explicit time evolution for the perturbation of potential temperature, we design a novel well-balanced finite difference WENO scheme for the conservative variables, which can be proven to be both asymptotic preserving and asymptotically accurate in the incompressible limit. Extensive numerical experiments were provided to validate these properties.