Asymptotic Preserving and Low Mach Number Accurate IMEX Finite Volume Schemes for the Euler Equations
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In this paper the design and analysis of a class of second order accurate IMEX finite volume schemes for the compressible Euler equations in the zero Mach number limit is presented. It is well known that in the zero Mach number limit fast acoustic waves are filtered out, which results in the incompressible Euler equations containing only slow advection waves. In order to account for the fast and slow waves, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components, respectively. The time discretisation is performed by an IMEX Runge-Kutta method, therein the stiff terms are treated implicitly and the non-stiff terms explicitly. In the space discretisation, a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. Both the time semi-discrete and space-time fully-discrete schemes are shown to be asymptotic preserving, i.e. both these schemes are consistent with the incompressible system in the zero Mach number limit, and their stability characteristics are independent of the Mach number. The numerical experiments confirm that the schemes achieve uniform second order convergence with respect to the Mach number. A notion of accuracy at low Mach numbers, termed as the asymptotic accuracy, is introduced in terms of the invariance of a well-prepared space of constant densities and divergence-free velocities. It is shown theoretically as well as numerically that the proposed schemes are asymptotically accurate.
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