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In this paper, the design and analysis of a class of second order accurateIMEX finite volume schemes for the compressible Euler equations in the zeroMach number limit is presented. In order to account for the fast and slowwaves, the nonlinear fluxes in the Euler equations are split into stiff andnon-stiff components, respectively. The time discretisation is performed by anIMEX Runge-Kutta method, therein the stiff terms are treated implicitly and thenon-stiff terms explicitly. In the space discretisation, a Rusanov-type centralflux is used for the non-stiff part, and simple central differencing for thestiff part. Both the time semi-discrete and space-time fully-discrete schemesare shown to be asymptotic preserving. The numerical experiments confirm thatthe schemes achieve uniform second order convergence with respect to the Machnumber. A notion of accuracy at low Mach numbers, termed as the asymptoticaccuracy, is introduced in terms of the invariance of a well-prepared space ofconstant densities and divergence-free velocities. The asymptotic accuracy isconcerned with the closeness of the compressible solution with that of itsincompressible counterpart in a low Mach number regime. It is showntheoretically as well as numerically that the proposed schemes areasymptotically accurate.

Asymptotic Preserving and Low Mach Number Accurate IMEX Finite Volume Schemes for the Isentropic Euler Equations