Truly multi-dimensional all-speed schemes for the Euler equations on Cartesian grids

TL;DR

This paper introduces a new multi-dimensional all-speed scheme for the Euler equations on Cartesian grids that improves stability and avoids free parameters by extending one-dimensional methods in a specific way.

Contribution

It proposes a novel multi-dimensional extension of one-dimensional low Mach number schemes that enhances stability without adding free parameters or arbitrary functions.

Findings

Significantly improves stability of Euler schemes at low Mach numbers

Maintains simplicity by avoiding free parameters

Applicable to various numerical methods like Lagrange Projection and relaxation solver

Abstract

Finite volume schemes often have difficulties to resolve the low Mach number (incompressible) limit of the Euler equations. Incompressibility is only non-trivial in multiple spatial dimensions. Low Mach fixes, however generally are applied to the one-dimensional method and the method is then used in a dimensionally split way. This often reduces its stability. Here, it is suggested to keep the one-dimensional method as it is, and only to extend it to multiple dimensions in a particular, all-speed way. This strategy is found to lead to much more stable numerical methods. Apart from the conceptually pleasing property of modifying the scheme only when it becomes necessary, the multi-dimensional all-speed extension also does not include any free parameters or arbitrary functions, which generally are difficult to choose, or might be problem dependent. The strategy is exemplified on a Lagrange Projection method and on a relaxation solver.