# American Institute of Mathematical Sciences

March  2012, 5(1): 155-184. doi: 10.3934/krm.2012.5.155

## Second order all speed method for the isentropic Euler equations

 1 Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, No. 800 Dong Chuan Road, Minhang, Shanghai 200240, China

Received  September 2011 Revised  September 2011 Published  January 2012

Standard hyperbolic solvers for the compressible Euler equations cause increasing approximation errors and have severe stability requirement in the low Mach number regime. It is desired to design numerical schemes that are suitable for all Mach numbers. A second order in both space and time all speed method is developed in this paper, which is an improvement of the semi-implicit framework proposed in [5].
The second order time discretization is based on second order Runge-Kutta method combined with Crank-Nicolson with some implicit terms. This semi-discrete framework is crucial to obtain second order convergence, as well as maintain the asymptotic preserving (AP) property. The AP property indicates that the right limit can be captured in the low Mach number regime. For the space discretization, the pressure term in the momentum equation is divided into two parts. Two subsystems are formed correspondingly, each using different space discretizations. One is discretized by Kurganov-Tadmor central scheme (KT), while the other one is reformulated into an elliptic equation. The proper subsystem division varies with time and the scheme becomes explicit when the time step is small enough.
Compared with previous semi-implicit method, this framework is simpler and natural, with only two linear elliptic equations needed to be solved for each time step. It maintains the AP property of the first order method in [5], improves accuracy and reduces the diffusivity significantly.
Citation: Min Tang. Second order all speed method for the isentropic Euler equations. Kinetic & Related Models, 2012, 5 (1) : 155-184. doi: 10.3934/krm.2012.5.155
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