An Energy Stable Well-balanced Scheme for the Barotropic Euler System with Gravity under the Anelastic Scaling

TL;DR

This work develops an energy-stable, structure-preserving, well-balanced, and asymptotic preserving finite-volume scheme for the barotropic Euler equations with gravity under the anelastic scaling. A velocity stabilization strategy in the convective fluxes, together with a careful discrete treatment of density at interfaces, yields discrete hydrostatic balance and robust stability across the full range of Mach/Froude numbers. The authors prove energy stability, well-balancing, weak consistency, and consistency with the anelastic limit, supported by a two-step solve (nonlinear elliptic density problem followed by velocity update) and a discrete relative-energy framework. Numerical experiments across 1D and 2D tests—including well-balancing, strong rarefaction, Sod-type shocks, perturbations of hydrostatic states, and a stationary vortex—confirm the scheme’s accuracy, positivity preservation, and AP behavior, making it suitable for multiscale atmospheric flow simulations.

Abstract

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.

Figures (10)

• Figure 1: Cross-sections of density and velocity profiles at time $T=0.1$ for the strong rarefaction problem.
• Figure 2: Density and velocity profiles for the Sod problem at time $T=0.2$.
• Figure 3: Comparison of the perturbations in density at time $T=0.25$ for $\varepsilon = 1$. (A) $\zeta = 10^{-3}$ and (B) $\zeta = 10^{-5}$. (C) Zoom of the plot in (B) for the initial perturbation and the well-balanced scheme.
• Figure 4: Comparison of the perturbations in density at time $T=0.25$. (A) $\varepsilon = 0.1$, (B) $\varepsilon = 0.01$ and (C) $\varepsilon = 0.001$. The amplitude of the perturbation is $\zeta = \varepsilon^2$. The non well-balanced scheme crashes beyond $\varepsilon=0.1$.
• Figure 5: Pseudo-colour plots and contours of the density perturbations obtained by (A) the non well-balanced explicit scheme and (B) the well-balanced scheme at time $T=0.05$ with $\varepsilon = 1$ and $\zeta=10^{-1}$.

Theorems & Definitions (52)

• Definition 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Remark 2.5
• Definition 2.6: Well-prepared initial data
• Theorem 2.7
• Corollary 2.8
• proof