Large‐scale simulation of shallow water waves via computation only on small staggered patches

Bunder, J. E.; Divahar, Jayaraman; Kevrekidis, Ioannis G.; Mattner, Trent W.; Roberts, Anthony John

doi:10.1002/fld.4915

Cited by 9 publications

(6 citation statements)

References 42 publications

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“…However, for problems with a shock existing in the initial state, like M1 and M3, the methods in this article will often be sufficient. Future research will look at the issues involved in the more difficult task of predicting shocks in multi-D by extending the multi-D patch scheme (Roberts et al 2014, Bunder et al 2019.…”

Section: Discussionmentioning

confidence: 99%

“…If the characteristic microscale length scales are much smaller than the macroscale, h H, then the patch scheme is much more efficient than computing over the full spatial domain. Further, for a variety of problems, and as commented in the Introduction, the patch scheme has been proven to be generally consistent to the underlying system to order H 2Γ (Roberts 2003, Roberts et al 2014, Bunder et al 2019.…”

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confidence: 87%

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A multiscale scheme accurately simulates macroscale shocks in an equation-free framework

Maclean,

Bunder,

Roberts

et al. 2020

Preprint

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Scientists and engineers often create accurate, trustworthy, computational simulation schemes-but all too often these are too computationally expensive to execute over the time or spatial domain of interest. The equation-free approach is to marry such trusted simulations to a framework for numerical macroscale reduction-the patch dynamics scheme. This article extends the patch scheme to scenarios in which the trusted simulation resolves abrupt state changes on the microscale that appear as shocks on the macroscale. Accurate simulation for problems in these scenarios requires extending the patch scheme by capturing the shock within a novel patch, and also modifying the patch coupling rules in the vicinity in order to maintain accuracy. With these two extensions to the patch scheme, straightforward arguments derive consistency conditions that match the usual order of accuracy for patch schemes. The new scheme is successfully tested on four archetypal problems. This technique will empower scientists and engineers to accurately and efficiently simulate, over large spatial domains, multiscale multiphysics systems that have rapid transition layers on the microscale.

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Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 87%

A multiscale scheme accurately simulates macroscale shocks in an equation-free framework

Maclean,

Bunder,

Roberts

et al. 2020

Preprint

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“…For a wide range of systems it has been proven that patch schemes with N patches generally possess an N -dimensional emergent slow manifold, a manifold parametrised by one 'macro-scale' variable per patch. These proofs apply to systems in multi-D space with either homogeneous micro-scale (Roberts et al 2014, Alotaibi et al 2018, Bunder et al 2021a or with heterogeneous micro-scale (Bunder et al , 2020. However, these proofs are predicated on the patches being stationary with identical spacing, which does not immediately hold for the moving patches of Section 3.…”

Section: Emergence and Consistency For Moving And Merging Patchesmentioning

confidence: 99%

“…For this u-system, the patches are stationary in ξ with identical spacing. Since the moving mesh pde (5) is diffusive, then for suitable dissipative pdes for u, the established emergence and consistency theories apply to the system for u, and hence to the moving patch scheme (Roberts et al 2014, Alotaibi et al 2018, Bunder et al 2021a, 2020.…”

Section: Emergence and Consistency For Moving And Merging Patchesmentioning

confidence: 99%

Adaptively detect and accurately resolve macro-scale shocks in an efficient Equation-Free multiscale simulation

Maclean¹,

Bunder²,

Kevrekidis³

et al. 2021

Preprint

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The Equation-Free approach to efficient multiscale numerical computation marries trusted micro-scale simulations to a framework for numerical macro-scale reduction-the patch dynamics scheme. A recent novel patch scheme empowered the Equation-Free approach to simulate systems containing shocks on the macro-scale. However, the scheme did not predict the formation of shocks accurately, and it could not simulate moving shocks. This article resolves both issues, as a first step in one spatial dimension, by embedding the Equation-Free, shock-resolving patch scheme within a classic framework for adaptive moving meshes. Our canonical micro-scale problems exhibit heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate many remarkable benefits from the moving patch scheme, including efficient and accurate macro-scale prediction despite the unknown macro-scale closure. Equation-free methods are here extended to simulate moving, forming and merging shocks without a priori knowledge of the existence or closure of the shocks. Whereas adaptive moving mesh equations are typically stiff, typically requiring small time-steps on the macro-scale, the moving

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“…Data-driven LES methods have successfully been developed in recent years, for example, by using neural networks to compute a variable eddy viscosity [4] to approximate a reference kinetic energy spectrum [33] or to model subgrid scale-scale forces [47]. Alternatively, approaches based on interpolation of small high-resolution patches of the spatial domain [10,9] and data-driven residual modeling via global basis functions [19] have also shown computational efficiency and accuracy in coarse-grained numerical solutions. Data assimilation provides an alternative method to achieving accurate coarse-grained results by combining predictions with real-time observations.…”

Section: Introductionmentioning

confidence: 99%

Computational Modeling for High-Fidelity Coarsening of Shallow Water Equations Based on Subgrid Data

Sagy

Luesink

Wimmer

et al. 2022

Multiscale Model. Simul.

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A resolution-independent data-driven stochastic parametrization method for subgrid-scale processes in coarsened fluid descriptions is proposed. The method enables the inclusion of high-fidelity data into the coarsened flow model, thereby enabling accurate simulations also with the coarser representation. The small-scale parametrization is introduced at the level of the Fourier coefficients of the coarsened numerical solution. It is designed to reproduce the kinetic energy spectra observed in high-fidelity data of the same system. The approach is based on a control feedback term reminiscent of continuous data assimilation. The method relies solely on the availability of high-fidelity data from a statistically steady state. No assumptions are made regarding the adopted discretization method or the selected coarser resolution. The performance of the method is assessed for the two-dimensional Euler equations on the sphere. Applying the method at two significantly coarser resolutions yields good results for the mean and variance of the Fourier coefficients. Stable and accurate large-scale dynamics can be simulated over long integration times.

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Large‐scale simulation of shallow water waves via computation only on small staggered patches

Cited by 9 publications

References 42 publications

A multiscale scheme accurately simulates macroscale shocks in an equation-free framework

A multiscale scheme accurately simulates macroscale shocks in an equation-free framework

Adaptively detect and accurately resolve macro-scale shocks in an efficient Equation-Free multiscale simulation

Computational Modeling for High-Fidelity Coarsening of Shallow Water Equations Based on Subgrid Data

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