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We consider two distinguished asymptotic limits of the Euler equations in agravitational field, namely the incompressible and Boussinesq limits. Boththese limits can be obtained as singular limits of the Euler equations underappropriate scaling of the Mach and Froude numbers. We propose and analyse anasymptotic preserving (AP) time discretisation for the numerical approximationof the Euler system in these asymptotic regimes. A key step in the constructionof the AP scheme is a semi-implicit discretisation of the fluxes and the sourceterm. The non-stiff convective terms are treated explicitly whereas the stiffpressure-gradient and source term are implicit. The implicit terms are combinedto get a nonlinear elliptic equation. We show that the overall scheme isconsistent with the respective limit system when the Mach number goes to zero.A linearised stability analysis confirms the $L^2$-stability of the proposedscheme. The results of numerical experiments validate the theoretical findings.

An Asymptotic Preserving Time Integrator for Low Mach Number Limits of the Euler Equations with Gravity