An Energy Stable Well‐Balanced Scheme for the Barotropic Euler System With Gravity Under the Anelastic Scaling

ABSTRACT We design and analyze an energy‐stable, structure‐preserving, well‐balanced, and asymptotic‐preserving (AP) scheme for the barotropic Euler system with gravity in the anelastic limit. The key to energy stability is the introduction of appropriate velocity shifts in the convective fluxes of mass and momenta. The semi‐implicit in time and finite volume in space fully‐discrete scheme supports the positivity of density and yields consistency with the weak solutions of the Euler system upon mesh refinement. The numerical scheme admits the discrete hydrostatic states as solutions and the stability of numerical solutions in terms of the relative energy leads to well‐balancing. The AP property of the scheme, that is, the boundedness of the mesh parameters to the Mach/Froude numbers and the scheme's asymptotic consistency with the anelastic Euler system is rigorously shown based on apriori energy estimates. The numerical scheme is resolved in two steps: by solving a non‐linear elliptic problem for the density and subsequent explicit computation of the velocity. Results from several benchmark case studies are presented to corroborate the proposed claims.