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In this paper, we propose a high order semi-implicit well-balanced finitedifference scheme for all Mach Euler equations with a gravitational sourceterm. To obtain the asymptotic preserving property, we start from theconservative form of full compressible Euler equations and add the evolutionequation of the perturbation of potential temperature. The resulting system isthen split into a (non-stiff) nonlinear low dynamic material wave to be treatedexplicitly, and (stiff) fast acoustic and gravity waves to be treatedimplicitly. With the aid of explicit time evolution for the perturbation ofpotential temperature, we design a novel well-balanced finite difference WENOscheme for the conservative variables, which can be proven to be bothasymptotic preserving and asymptotically accurate in the incompressible limit.Extensive numerical experiments were provided to validate these properties.

High order asymptotic preserving well-balanced finite difference WENO schemes for all Mach full Euler equations with gravity