The solution of the compressible Euler equations at low Mach numbers using a stabilized finite element algorithm

Abstract

We present a streamline-upwind/Petrov–Galerkin (SUPG) algorithm for the solution of the compressible Euler equations at low Mach numbers. The Euler equations are written in terms of entropy variables which result in Jacobian matrices which are symmetric. We note that, in the low Mach number limit, the SUPG method with the standard choices for the stabilization matrix fail to provide adequate stabilization. This results in a degradation of the solution accuracy. We propose a stabilization matrix which incorporates dimensional-scaling arguments and which exhibits the appropriate behavior for low Mach numbers. To guide in the derivation of the new stabilization matrix, the non-dimensionalized equations are transformed to a new set of variables that converge, when the characteristic Mach number tends to zero, to the incompressible velocity and pressure variables. The resulting algorithm is capable of accurately computing external flows with free stream Mach numbers as low as 10 −3.