A high‐order WENO‐limited finite‐volume algorithm for atmospheric flow using the ADER‐differential transform time discretization

Norman, Matthew R.

doi:10.1002/qj.3989

Cited by 11 publications

(20 citation statements)

References 68 publications

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“…We consider a two-dimensional thermal collision of two hot and cold thermals as a sample problem to demonstrate the modeling framework [29]. The colliding thermals create strong discontinuities, strong winds, and significant turbulent regimes as the flow evolves in time.…”

Section: Fluid Flow Problemmentioning

confidence: 99%

“…The temporal regimes consisting of both anti-dissipative and dissipative effects makes the colliding thermal test case a useful model problem to test the capability of the current framework to capture these effects. Further details of the test case, solver, and numerical schemes can be found in Norman [29]. Figure 2: Time evolution of the colliding thermal problem from initial condition to a turbulent state portrayed using potential temperature θ .…”

Section: Fluid Flow Problemmentioning

confidence: 99%

“…We train the model in Python environment using PyTorch and then deploy the model in the C++ solver [29] with GPU acceleration for production usage. The "NAdam" optimizer is used for optimizing the weights of the NNs [36][37][38].…”

Section: Training and Validation Of Modelmentioning

confidence: 99%

See 2 more Smart Citations

Surrogate modeling of subgrid-scale effects in idealized atmospheric flows: A deep learned approach using high-resolution simulation data

Meena

Norman

Hall

et al. 2023

Preprint

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Section: Fluid Flow Problemmentioning

confidence: 99%

Section: Fluid Flow Problemmentioning

confidence: 99%

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Surrogate modeling of subgrid-scale effects in idealized atmospheric flows: A deep learned approach using high-resolution simulation data

Meena

Norman

Hall

et al. 2023

Preprint

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“…Then, the time derivatives in (10) are written in terms of the u and t derivatives of F by using the chain rule…”

Section: Ordinary Differential Equationsmentioning

confidence: 99%

“…Arbitrary derivatives (ADER) time-stepping methods involving space-time basis functions also belong to this category. Norman [10] develops a high-order WENO-limited finite-volume algorithm for modeling atmospheric flow using the ADER-differential transform time discretization. Models like MITgcm [11] use direct space-time methods for modeling non-linear advection and other physical phenomena in the ocean.…”

Section: Introductionmentioning

confidence: 99%

On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

Bishnu¹,

Peterson²,

Quaife³

2021

Preprint

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In this work, we determine the full expression of the local truncation error of hyperbolic partial differential equations (PDEs) on a uniform mesh. At a time horizon, the global truncation error, which is one order of the time step less than its local counterpart, assumes the formwhere ∆x and ∆t denote the cell width and the time step size, and α and β represent the orders of the spatial and temporal discretizations. If we are employing a stable numerical scheme and the global solution error is of the same order of accuracy as the global truncation error, we make the following observations in the asymptotic regime, where the truncation error is dominated by the powers of ∆x and ∆t rather than their coefficients. Assuming that we reach the asymptotic regime before the machine precision error takes over, (a) the order of convergence of stable numerical solutions of hyperbolic PDEs at constant ratio of ∆t to ∆x is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. We have tested our theory against numerical methods employing Method of Lines and not against ones that treat space and time together, and we have not taken into consideration the reduction in the spatial and temporal orders of accuracy resulting from slope-limiting monotonicity-preserving strategies commonly applied to finite volume methods. Otherwise, our theory applies to any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method. If the PDE is reduced to an ordinary differential equation (ODE) by specifying the spatial gradients of the dependent variable and the coefficients and the source terms to be zero, then the standard local truncation error of the ODE is recovered. We perform the analysis with generic and specific hyperbolic PDEs using the symbolic algebra package SymPy, and conduct a number of numerical experiments to demonstrate our theoretical findings.

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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.

show abstract

A high‐order WENO‐limited finite‐volume algorithm for atmospheric flow using the ADER‐differential transform time discretization

Cited by 11 publications

References 68 publications

Surrogate modeling of subgrid-scale effects in idealized atmospheric flows: A deep learned approach using high-resolution simulation data

Surrogate modeling of subgrid-scale effects in idealized atmospheric flows: A deep learned approach using high-resolution simulation data

On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

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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