We design and analyse 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 the 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, i.e. the boundedness of the mesh parameters with respect to the Mach/Froude numbers and the scheme's asymptotic consistency with the anelastic Euler system is rigorously shown on the basis of apriori energy estimates. The numerical scheme is resolved in two steps: by solving a non-linear elliptic problem for the density and a subsequent explicit computation of the velocity. Results from several benchmark case studies are presented to corroborate the proposed claims.

翻译：我们设计并分析了一种适用于滞弹性极限下重力场中正压欧拉系统的能量稳定、结构保持、良好平衡且渐近保持（AP）格式。实现能量稳定性的关键在于：在质量通量与动量通量的对流项中引入适当的速度偏移。该时间半隐式与空间有限体积的全离散格式支持密度正定性，并在网格细化下与欧拉系统弱解保持一致性。该数值格式将离散静力平衡状态作为解接受，且基于相对能量的数值解稳定性导致了良好平衡特性。基于先验能量估计，严格证明了格式的AP性质（即网格参数关于马赫数/弗劳德数的有界性）及其与滞弹性欧拉系统的渐近一致性。该数值方案通过两步求解：首先求解密度的非线性椭圆问题，随后显式计算速度。最后通过多个基准算例的数值结果验证了上述理论断言。