Semi-Implicit Extension of a Godunov-Type Scheme Based on Low Mach Number Asymptotics I: One-dimensional Flow,

Abstract

A single scale, multiple space scale asymptotic analysis provides detailed insight into the low Mach number limit behavior of solutions of the compressible Euler equations. We use the asymptotics as a guideline for developing a low Mach number extension of an explicit higher order shock-capturing scheme. This semi-implicit scheme involves multiple pressure variables, large scale differencing and averaging procedures that are discretized versions of standard operations in multiple scales asymptotic analysis. Advection and acoustic wave propagation are discretized explicitly and upwind and only one scalar elliptic equation is to be solved implicitly at each time step. This equation is a pressure correction equation for incompressible flows when the Mach number is zero. In the low Mach number limit, the time step is restricted by a Courant number based essentially on the maximum flow velocity. For moderate and large Mach numbers the scheme reduces to the underlying explicit higher order shock capturing algorithm.