Runge-Kutta Discontinuous Local Evolution Galerkin Methods for the Shallow Water Equations on the Cubed-Sphere Grid

Kuang, Yangyu; Wu, Kailiang; Tang, Huazhong

doi:10.4208/nmtma.2017.s09

Cited by 3 publications

(1 citation statement)

References 61 publications

(111 reference statements)

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“…Numerical simulation is an effective tool to solve them and a great variety of numerical methods are available in the literature, e.g. [4,31,37,47,48,52,53,54] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics

Duan,

Tang

2020

Preprint

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This paper develops the high-order accurate entropy stable (ES) finite difference schemes for the shallow water magnetohydrodynamic (SWMHD) equations. They are built on the numerical approximation of the modified SWMHD equations with the Janhunen source term. First, the second-order accurate well-balanced semi-discrete entropy conservative (EC) schemes are constructed, satisfying the entropy identity for the given convex entropy function and preserving the steady states of the lake at rest (with zero magnetic field). The key is to match both discretizations for the fluxes and the non-flat river bed bottom and Janhunen source terms, and to find the affordable EC fluxes of the second-order EC schemes. Next, by using the second-order EC schemes as building block, high-order accurate well-balanced semi-discrete EC schemes are proposed. Then, the high-order accurate well-balanced semidiscrete ES schemes are derived by adding a suitable dissipation term to the EC scheme with the WENO reconstruction of the scaled entropy variables in order to suppress the numerical oscillations of the EC schemes. After that, the semi-discrete schemes are integrated in time by using the high-order strong stability preserving explicit Runge-Kutta schemes to obtain the fully-discrete high-order well-balanced schemes. The ES property of the Lax-Friedrichs flux is also proved and then the positivity-preserving ES schemes are studied by using the positivity-preserving flux limiter. Finally, extensive numerical tests are conducted to validate the accuracy, the well-balanced, ES and positivity-preserving properties, and the ability to capture discontinuities of our schemes.

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“…Numerical simulation is an effective tool to solve them and a great variety of numerical methods are available in the literature, e.g. [4,31,37,47,48,52,53,54] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics

Duan,

Tang

2020

Preprint

Self Cite

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High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography

Zhang,

Duan,

Tang

2023

Journal of Computational Physics

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High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations

Gao

2017

Commun. Comput. Phys.

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In this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

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Runge-Kutta Discontinuous Local Evolution Galerkin Methods for the Shallow Water Equations on the Cubed-Sphere Grid

Cited by 3 publications

References 61 publications

High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics

High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics

High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography

High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations

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