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An all speed scheme for the Isentropic Euler equation is presented in thispaper. When the Mach number tends to zero, the compressible Euler equationconverges to its incompressible counterpart, in which the density becomes aconstant. Increasing approximation errors and severe stability constraints arethe main difficulty in the low Mach regime. The key idea of our all speedscheme is the special semi-implicit time discretization, in which the low Machnumber stiff term is divided into two parts, one being treated explicitly andthe other one implicitly. Moreover, the flux of the density equation is alsotreated implicitly and an elliptic type equation is derived to obtain thedensity. In this way, the correct limit can be captured without requesting themesh size and time step to be smaller than the Mach number. Compared withprevious semi-implicit methods, nonphysical oscillations can be suppressed. Wedevelop this semi-implicit time discretization in the framework of a firstorder local Lax-Friedrich (LLF) scheme and numerical tests are displayed todemonstrate its performances.

All speed scheme for the low mach number limit of the Isentropic Euler equation