High-order Simulation of Nearly Hydrostatic Atmospheric Flows

Abstract

Accurate simulation of atmospheric flows is challenging. Low-Mach-number dynamics often need to be resolved in the presence of a dominant, nearly hydrostatic background state. Conventional upwind numerical fluxes can introduce excessive acoustic-speed-based dissipation in the low-Mach regime. Discretization errors in the balance law can also generate spurious motion. This work investigates these two issues within a high-order Direct Flux Reconstruction (DFR) framework coupled with a third-order exponential time integrator for the 3D compressible Euler equations with gravity. Low-Mach-number accuracy is addressed using a modified AUSM${}^{+}$-up flux, denoted AUSM${}^{+}$-upH. Preservation of the background equilibrium is improved through a deviation-based formulation. The resulting approach is assessed using a hierarchy of idealized atmospheric test cases. The results show that AUSM${}^{+}$-upH preserves low-Mach flow structures more effectively than the more dissipative Rusanov flux. However, reduced numerical dissipation also makes AUSM${}^{+}$-upH more sensitive to residual errors in the discrete hydrostatic balance. The deviation formulation substantially suppresses these errors by several orders of magnitude, with the strongest improvement observed for flows over orography. Overall, AUSM${}^{+}$-upH exhibits greater robustness when combined with the deviation formulation across the atmospheric test cases considered.