Nonlinear diffusion‐limited two‐dimensional colliding‐plume simulations with very high‐order numerical approximations

Abstract

AbstractAtmospheric numerical models play a crucial role in operational weather forecasting, as well as in improving our understanding of atmospheric dynamics via research studies. Maximising their accuracy is of paramount importance. Use of advective flux schemes greater than O7 in atmospheric models is largely undocumented, with no studies considering O3–O17 fluxes with formal accuracy‐preserving high‐order interpolation, pressure gradient/divergence, and subgrid‐scale (SGS) turbulent fluxes. Higher‐order numerical approximations can reduce truncation, amplitude and phase errors, and potentially improve model accuracy and effective resolution. Here, simulations are presented using very‐high‐order O3–O17 fluxes, with/without high‐order O2–O18 Lagrangian interpolations, pressure gradient/divergence approximations, and SGS turbulent fluxes for a two‐dimensional, highly‐viscous (Re ∼ 100) diffusion‐limited, nonlinear colliding‐plumes problem using 25–200 m spatial resolutions. The highest‐order flux schemes coupled with higher‐order interpolations, pressure gradient/divergence and SGS flux approximations produced the best solutions, with higher‐order fluxes and interpolations being most important. Overall solution convergence of order about O1–O2 with mode‐split (fast sound/slow advective waves) O3 Runge–Kutta temporal schemes was negatively impacted by order at most O1 temporal convergence with SGS fluxes, divergence damping, and especially spatial filters, compared to about order O3 convergence with these inactivated. While very‐high‐order schemes were shown to improve solution accuracy, few cost‐effective higher‐order highly‐viscous test problem solutions (higher order vs finer resolution) were found using theoretical floating‐point operations (FPO) with Courant‐Friedrichs‐Lewy (CFL)‐limited or constant stable Courant number‐based time steps. However, employing central processing unit (CPU) time, rather than FPOs, demonstrated there was reduced computational burden using higher‐order approximations. We conclude that O9–O17 flux schemes with or without high‐order (≥O4) interpolations, pressure gradient/divergence approximations, and SGS fluxes can improve atmospheric model solution accuracy, without prohibitive computation costs, compared to O3–O7 flux with O2 interpolations, pressure gradient/divergence approximations, and SGS fluxes.