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

Two-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 Delta x = Delta z = 25-166.66 & mldr; 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 (= 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.