Commun. Comput. Phys., 10 (2011), pp. 1-31. All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations Pierre Degond 1, Min Tang 1*1 Institute of Mathematics of Toulouse, Universite Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France. Received 21 July 2009; Accepted (in revised version) 21 June 2010 Available online 7 February 2011 doi:10.4208/cicp.210709.210610a Abstract An all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods [11, 13, 29], firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests are displayed to demonstrate its performances. AMS subject classifications: 65M06, 65Z05, 76N99, 76L05 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Low Mach number, Isentropic Euler equations, compressible flow, incompressible limit, asymptotic preserving, Rusanov scheme. *Corresponding author. Email: degond@mip.ups-tlse.fr (P. Degond), tangmin1002@gmail.com (M. Tang)