A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows

Xiong, Tao; Qiu, Jing-Mei; Xu, Zhengfu

doi:10.1016/j.jcp.2013.06.026

Cited by 49 publications

(71 citation statements)

References 18 publications

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“…In the above two scenarios, the expansions result in similar formulas to those given in the second part of the proof of Theorem 3.2 in [19], and satisfy the exact same relations as those in that proof. Using the same argument as the proof of [19], it can be shown one of the relations u(…”

Section: Case (2)supporting

confidence: 66%

“…The main advantage of this new parametrized limiter is that the designed order of accuracy of the base WENO/DG schemes are maintained without excessively restricting the CFL. Later in [19], Xiong et al improved the CFL constraint and reduced computational cost by applying the parametrized MPP flux limiter to the final RK stage only. It was also proven in [19] that the parametrized MPP flux limiter can maintain up to third order accuracy in space and time for one-dimensional nonlinear scalar conservation laws on uniform meshes.…”

Section: Introductionmentioning

confidence: 99%

“…Later in [19], Xiong et al improved the CFL constraint and reduced computational cost by applying the parametrized MPP flux limiter to the final RK stage only. It was also proven in [19] that the parametrized MPP flux limiter can maintain up to third order accuracy in space and time for one-dimensional nonlinear scalar conservation laws on uniform meshes. This limiter is also extended to high order PP finite difference WENO schemes for the compressible Euler system in [18] and high order MPP finite volume WENO methods for scalar convection-dominated problems on uniform meshes [21].…”

Section: Introductionmentioning

confidence: 99%

“…

In this paper, we generalize the maximum-principle-preserving (MPP) flux limiting technique developed by Xu (2013) [20] to a class of high order finite volume weighted essentially non-oscillatory (WENO) schemes for scalar conservation laws and the compressible Euler system on unstructured meshes in one and two dimensions. The key idea of this parameterized limiting technique is to limit the high order numerical flux with a first order flux which preserves the MPP or positivity-preserving (PP) property.

…”

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confidence: 99%

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High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

Christlieb

Liu

Tang

et al. 2015

Journal of Computational Physics

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Section: Case (2)supporting

confidence: 66%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

Christlieb

Liu

Tang

et al. 2015

Journal of Computational Physics

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“…In addition, some more recently proposed maximum-principle-preserving algorithms can assure the non-negativity of high-order transport models. A flux correction algorithm was proposed in Xiong et al (2013) for the high-order Runge-Kutta RK-WENO schemes to guarantee the non-negative solution by imposing the restrictions on the deviations between high-order and low-order fluxes. This algorithm can be applied in a one-dimensional multimoment model following the similar numerical procedure described hereafter by modifying the PVs according to the values of the corrected fluxes.…”

Section: Introductionmentioning

confidence: 99%

A note on non‐negativity correction for a multimoment finite‐volume transport model with WENO limiter

Shen

Chen

et al. 2019

Quart J Royal Meteoro Soc

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An oscillation‐less multimoment global transport model was proposed by Tang et al. in 2018 by introducing a slope limiter based on the weighted essentially non‐oscillatory (WENO) concept. The spurious oscillations around the discontinuities and strong gradients can be effectively removed and the model preserves the fourth‐order accuracy in spherical geometry for smooth solutions. However, the WENO limiter does not strictly guarantee the non‐negativity of numerical solution and the resulting model will violate the physical principle due to the existence of the nonphysical negative undershoots. As the numerical fluxes, which determine the time tendency of the volume‐integrated average, are evaluated by the point values (PVs) defined along the cell boundaries in the multimoment finite‐volume schemes, the non‐negativity preserving model can be accomplished by modifying those PVs to implement the corrections on the numerical fluxes proposed in the positive definite flux‐corrected transport (FCT) method by Smolarkiewicz in 1989. In this study, a non‐negativity correction algorithm is proposed for a global transport model using the MM‐FVM_WENO scheme and verified by simulating the widely used benchmark tests.

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Impulsive fishery resource transporting strategies based on an open‐ended stochastic growth model having a latent variable

Yoshioka

Tanaka

Aranishi

et al. 2021

Math Methods in App Sciences

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In inland fisheries, transporting fishery resource individuals from a habitat to spatially apart habitat(s) has recently been considered for fisheries stock management in the natural environment. However, its mathematical optimization, especially finding when and how much of the population should be transported, is still a fundamental unresolved issue. We propose a new impulse control framework to tackle this issue based on a simple but new stochastic growth model of individual fishes. The novel growth model governing individuals' body weights uses a Wright-Fisher model as a latent driver to reproduce plausible growth dynamics. The optimization problem is formulated as an impulse control problem of a cost-benefit functional constrained by a degenerate parabolic Fokker-Planck equation of the stochastic growth dynamics.Because the growth dynamics have an observable variable and an unobservable variable (a variable difficult or impossible to observe), we consider both full-information and partial-information cases. The latter is more involved but more realistic because of not explicitly using the unobservable variable in designing the controls. In both cases, resolving an optimization problem reduces to solving the associated Fokker-Planck and its adjoint equations, the latter being non-trivial. We present a derivation procedure of the adjoint equation and its internal boundary conditions in time to efficiently derive the optimal transporting strategy.We finally provide a demonstrative computational example of a transporting problem of Ayu sweetfish (Plecoglossus altivelis altivelis) based on the latest real data set.

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A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows

Cited by 49 publications

References 18 publications

High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

A note on non‐negativity correction for a multimoment finite‐volume transport model with WENO limiter

Impulsive fishery resource transporting strategies based on an open‐ended stochastic growth model having a latent variable

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