Very high order WENO schemes using efficient smoothness indicators

Wu, Conghai; Wu, Ling; Li, Hu; Zhang, Shuhai

doi:10.1016/j.jcp.2021.110158

Cited by 17 publications

(11 citation statements)

References 46 publications

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“…Results from a comprehensive suite of simulations of a dry nonlinear two‐dimensional test problem are presented to demonstrate the impact of very‐high‐order finite difference using O3, O5, O9, O13 and O17 upwind‐biased advection/flux approximations (same approximation orders were also used by G09 and Wu et al, 2021 for various WENO flux schemes), coupled with comparable even‐order Lagrangian interpolations for information required at off‐grid point locations and even‐order staggered pressure gradient/divergence approximations (one order higher for odd‐order schemes; e.g., for O17 advection, O18 interpolation/pressure gradient/divergence is used). A summary of the grid resolutions (200, 166.66… [hereafter 166.67], 133.33… [133.33], 100, 66.66… [66.67], 50, 33.33… [33.33], 25 m) and domain parameters used in this study are in Table 1, while orders of accuracy for fluxes, interpolations, pressure gradient/divergence, diffusion, and spatial filters are given in Table 2.…”

Section: Resultsmentioning

confidence: 99%

“…Results from a comprehensive suite of simulations of a dry nonlinear two-dimensional test problem are presented to demonstrate the impact of very-high-order finite difference using O3, O5, O9, O13 and O17 upwind-biased advection/flux approximations (same approximation orders were also used by G09 and Wu et al, 2021 for various WENO flux schemes), coupled with comparable even-order Lagrangian interpolations for information required at off-grid point locations and even-order staggered pressure gradient/divergence approximations (one order higher for odd-order schemes; e.g., for O17 Note: U t is the added wind, which is U t = 0 m⋅s −1 for all experiments, except Set B which has U t = −20 m⋅s −1 . The meaning of Oc is comparable order (one order higher than used for odd-order advective flux); for example, in Set A, O3 fluxes are coupled with comparable-order (next order higher) O4 interpolations, O4 pressure gradient/divergence, O18 spatial filter, and O2 subgrid-scale (SGS) turbulent fluxes (diffusion).…”

Section: Two-dimensional Colliding Plumesmentioning

confidence: 99%

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

Straka,

Williams,

Kanak

2024

Quart J Royal Meteoro Soc

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Atmospheric numerical models play a crucial role in operational weather forecasting, as well as improving our understanding of atmospheric dynamics via research studies. Maximising their accuracy is of paramount importance. Use of >O7 flux schemes in atmospheric models is largely undocumented, with no studies considering O3–17 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 O3–17 fluxes, with / without high order O2–18 Lagrangian interpolations, pressure gradient / divergence approximations, and SGS turbulent fluxes for a 2D, 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 ~O1–2 with mode‐split (fast sound / slow advective waves) O3 Runge–Kutta temporal schemes was negatively impacted by ≤O1 temporal convergence with SGS fluxes, divergence damping, and especially spatial filters, compared to ~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 versus finer resolution) were found using theoretical floating‐point operations (FPO) with CFL‐limited or constant stable Courant number‐based time steps. However, employing CPU time, rather than FPOs, demonstrated there was reduced computational burden using higher order approximations. We conclude that O9–17 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 O3–7 flux with O2 interpolations, pressure gradient / divergence approximations, and SGS fluxes.This article is protected by copyright. All rights reserved.

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Section: Resultsmentioning

confidence: 99%

Section: Two-dimensional Colliding Plumesmentioning

confidence: 99%

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

Straka,

Williams,

Kanak

2024

Quart J Royal Meteoro Soc

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“…This enables the use of the same stencil and numerical scheme as those used in the interior grids, and also facilitates the communication in the message passing interface (MPI) parallelisation. At the ghost nodes, the conserved variables are assigned based on the known values at the interior nodes according to specific physical boundary conditions (Fu 2021;Wu et al 2021). It should be pointed out that for the outflow boundary condition, the standard zero-gradient extrapolation may result in spurious nonphysical reflection of the outgoing waves (Motheau, Almgren & Bell 2017).…”

Section: Methodsmentioning

confidence: 99%

“…At the ghost nodes, the conserved variables are assigned based on the known values at the interior nodes according to specific physical boundary conditions (Fu 2021; Wu et al. 2021). It should be pointed out that for the outflow boundary condition, the standard zero-gradient extrapolation may result in spurious nonphysical reflection of the outgoing waves (Motheau, Almgren & Bell 2017).…”

Section: Methodsmentioning

confidence: 99%

Mixing and inter-scale energy transfer in Richtmyer–Meshkov turbulence

Zhou,

Ding,

Cheng

2024

J. Fluid Mech.

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This work investigates the compressible turbulence induced by Richtmyer–Meshkov (RM) instability using high-resolution Navier–Stokes simulations. Special attention is paid to the characteristics of RM turbulence including the mixing width growth, the turbulent kinetic energy (TKE) decay, the mixing degree, inhomogeneity and anisotropy. Three distinct initial perturbation spectra are designed at the interface to reveal the initial condition imprint on the RM turbulence. Results show that cases with initial large-scale perturbations present a stronger imprint on statistical characteristics and also a quicker growth of mixing width, whereas cases with small-scale perturbations present a faster TKE decay, greater mixing level, higher isotropy and homogeneity. A thorough analysis on the inter-scale energy transfer in RM turbulence is also presented with the coarse-graining approach that exposes the two subfilter-scale (SFS) energy fluxes (i.e. deformation work and baropycnal work). A strong correlation between the nonlinear model of baropycnal work (Fluids, 4(2), 2019) and the simulation results is confirmed for the first time, demonstrating its potential in modelling the RM turbulence. Two primary mechanisms of baropycnal work (the straining and baroclinic generation processes) are explored with this nonlinear model. The evolutions of two SFS energy fluxes exhibit distinct behaviours at various filter scales, in different flow regions and under various flow motions (strain and rotation). It is found that all three cases share the common inter-scale energy transfer dynamics, which is important for modelling the RM turbulence.

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“…Much of the computational cost of WENO schemes is associated with computing the smoothness indicators (e.g. S97; G09; Wu et al ., 2020 W20; Wu et al ., 2021 W21). However, W20 and W21 recently described simpler, very accurate smoothness indicators (exact for sine‐waves), which are much more efficient than traditional ≥O7 WENO schemes.…”

Section: Introductionmentioning

confidence: 99%

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

Straka,

Kanak,

Williams

2023

Quart J Royal Meteoro Soc

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A two‐dimensional, non‐linear 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 modelsThis article is protected by copyright. All rights reserved.

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Very high order WENO schemes using efficient smoothness indicators

Cited by 17 publications

References 46 publications

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

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

Mixing and inter-scale energy transfer in Richtmyer–Meshkov turbulence

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

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