All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations (Communications in Computational Physics, 2011)
An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations convergeto their incompressible counterpart, in which the density becomes a constant. Increasingapproximation errors and severe stability constraints are the main difficultyin the low Mach regime. The key idea of our all speed scheme is the special semi-implicittime discretization, in which the low Mach number stiff term is divided intotwo parts, one being treated explicitly and the other one implicitly. Moreover, the fluxof the density equation is also treated implicitly and an elliptic type equation is derivedto obtain the density. In this way, the correct limit can be captured without requestingthe mesh size and time step to be smaller than the Mach number. Compared withprevious semi-implicit methods [11,13,29], firstly, nonphysical oscillations can be suppressedby choosing proper parameter, besides, only a linear elliptic equation needs tobe solved implicitly which reduces much computational cost. We develop this semi-implicittime discretization in the framework of a first order Local Lax-Friedrichs (orRusanov) scheme and numerical tests are displayed to demonstrate its performances.