Comparison of very high order (O3–18) flux and Crowley flux, and (O3–17) WENO flux schemes with a 2D nonlinear test problem

Abstract

AbstractTwo‐dimensional, nonlinear diffusion‐limited colliding plumes simulations were used to demonstrate the improved solution accuracy with very high order O9–18 flux schemes, including upwind‐biased and even‐centred constant grid flux and Crowley constant grid flux schemes, and odd order weighted essentially non‐oscillatory (WENO) flux schemes, along with variations and hybrids of these. All schemes were coupled with comparably high order even‐centred Lagrangian interpolations and pressure gradient/divergence approximations, and O18 spatial filtering. Subgrid‐scale (SGS) turbulent flux calculations, with a constant eddy‐mixing coefficient, were made with O2 spatial approximations (O4–20 accurate SGS turbulent fluxes had little impact). Using a range of resolutions from Δx = Δz = 25–166.66… m for all schemes comparisons against an O17 flux, 25 m resolution reference solution showed solutions made with ≥O9 fluxes produced (often substantially) improved solutions, both visually and usually objectively, compared to solutions produced with lower order (<O9/10) fluxes, especially at intermediate resolutions (33.33–100 m). Expectedly, odd order solutions were increasingly damped as accuracy was decreased, especially from O9 to O3, especially for WENO solutions, while even order solutions were increasingly contaminated with dispersion and aliasing errors as accuracy was decreased, especially from O10 to O4. Odd order schemes also produced better solutions than even order schemes for <O9/10 fluxes, while the highest order (≥O13/14) schemes produced the best solutions, for any given resolution. Even order flux and Crowley flux (WENO) solutions were the least (most) computationally expensive, based on either floating‐point operations (FPO) or CPU times. Efficient WENO‐Sine and proposed hybrid Crowley‐WENO(‐Sine) schemes required fewer FPOs to produce more accurate solutions than traditional WENO schemes. We are encouraged by the often much improved visual and objective accuracy of very high order (≥O9) fluxes in simulations of a complex problem, and encourage further testing in numerical weather prediction models.