Investigating Inherent Numerical Stabilization for the Moist, Compressible, Non‐Hydrostatic Euler Equations on Collocated Grids

Abstract

Abstract This study investigates inherent numerical dissipation due to upwind fluxes and reconstruction strategies for collocated Finite‐Volume integration of the Euler equations. Idealized supercell simulations are used without any explicit dissipation. Flux terms are split into: mass flux, pressure, and advected quantities. They are computed with the following upwind strategies: central, advectively upwind, and acoustically upwind. This is performed for third and ninth‐order‐accurate reconstructions with and without Weighted Essentially Non‐Oscillatory limiting. Acoustic‐only upwinding for pressure and mass flux terms and advective‐only upwinding for advected quantities is the most flexible simplification found. It reduces data movement and computations. Assuming a constant speed of sound in acoustic upwinding gives similar results to using the true speed of sound. Dissipation from upwind adapts automatically to grid spacing, time step, reconstruction accuracy, and flow smoothness. While stability is maintained even at 21st‐order spatial accuracy, there is a limit to the spatial order of accuracy for which upwinding alone can create a realizable solution in the conditions of this study. Convex combinations of upwind and central solutions for flux terms also reduced dissipation, but as the central proportion grows, solutions become physically unrealizable. The range of length scales of the kinetic energy spectra can be extended along k−5/3 to smaller spatial scales by reducing dissipation either with higher‐order reconstructions or using convex combinations of upwind and central fluxes. However, not all extensions of the length scale range along k−5/3 exhibit physically realizable solutions, even though the spectra appear to be physical.

Keywords

• WENO
• upwind
• finite‐volume
• collocated
• stability
• supercell