All-speed numerical methods for the Euler equations via a sequential explicit time integration

TL;DR

This paper introduces an all-speed explicit numerical method for the Euler equations that reduces diffusion at low Mach numbers by combining centered discretizations with a leap-frog time integration inspired by staggered grid schemes.

Contribution

It develops a new all-speed explicit scheme for Euler equations using a novel variable staggering approach inspired by staggered grid methods, improving stability and accuracy.

Findings

Reduces excessive diffusion in low Mach regimes

Provides new collocated schemes inspired by Yee's method

Enhances stability through a novel variable staggering approach

Abstract

This paper presents a new strategy to deal with the excessive diffusion that standard finite volume methods for compressible Euler equations display in the limit of low Mach number. The strategy can be understood as using centered discretizations for the acoustic part of the Euler equations and stabilizing them with a leap-frog-type ("sequential explicit") time integration, a fully explicit method. This time integration takes inspiration from time-explicit staggered grid numerical methods. In this way, advantages of staggered methods carry over to collocated methods. The paper provides a number of new collocated schemes for linear acoustic/Maxwell equations that are inspired by the Yee scheme. They are then extended to an all-speed method for the full Euler equations on Cartesian grids. By taking the opposite view and taking inspiration from collocated methods, the paper also suggests a new way of staggering the variables which increases the stability as compared to the traditional Yee scheme.