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

• Published in: Journal of advances in modeling earth systems Vol. 15; no. 10
• Main Authors: Norman, Matthew R., Eldred, Christopher, Gopalakrishnan Meena, Muralikrishnan
• Format: Journal Article
• Language: English
• Published: Washington John Wiley & Sons, Inc 01.10.2023; American Geophysical Union (AGU)
• Subjects: Accuracy; Atmospheric models; collocated; Energy spectra; Eulers equations; finite‐volume; Fluctuations; Kinetic energy; Mass flux; Preferences; Spectra; stability; supercell; Turbulence models; upwind; Variables; Viscosity; WENO
• ISSN: 1942-2466; 1942-2466
• DOI: 10.1029/2023MS003732

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. Integrating the Euler equations that govern certain regimes of fluid motion forward in time must be performed carefully. Subtle choices in the numerical algorithm can lead to instability or solutions that are unphysical. “Upwinding” is an algorithmic choice where the developer prefers information that comes from the upwind direction, and it is known to lead to smoother and more stable solutions in general. This study explores the effects on stability and physical realism of using different upwind preferences for different variables in the fluxes of fluid properties between adjacent cells in a discrete 3‐D grid. Sound waves are a part of the equations of atmospheric motion. They propagate through time on the edge of an expanding sphere, which makes determining a single “upwind” direction more complex. One can transform the equations into an equivalent form called “characteristic” form, though, that simplifies this definition. That transformation couples the variables together tightly and requires extra computations and data movement across a large parallel machine. This paper shows that there are simplifications to this fully upwind characteristic approach that are just as physically realistic and stable but reduce the number of computations and volume of data transfers significantly. There are simplifications to a fully upwind Riemann solver that require fewer computations and less data movement Upwinding according to advective versus acoustic dynamics makes a large difference in stability One can use convex combinations of central and upwind fluxes to obtain physically realizable solutions with less dissipation