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A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations

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  • Gassner, Gregor J.
  • Winters, Andrew R.
  • Kopriva, David A.

Abstract

In this work, we design an arbitrary high order accurate nodal discontinuous Galerkin spectral element type method for the one dimensional shallow water equations. The novel method uses a skew-symmetric formulation of the continuous problem. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, we show that combined with a special numerical interface flux function, the method exactly preserves the entropy, which is also the total energy for the shallow water equations. Finally, we prove that the surface fluxes, the skew-symmetric volume integrals, and the source term are well balanced. Numerical tests are performed to demonstrate the theoretical findings.

Suggested Citation

  • Gassner, Gregor J. & Winters, Andrew R. & Kopriva, David A., 2016. "A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations," Applied Mathematics and Computation, Elsevier, vol. 272(P2), pages 291-308.
    • Handle: RePEc:eee:apmaco:v:272:y:2016:i:p2:p:291-308
    DOI: 10.1016/j.amc.2015.07.014
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    Cited by:

    1. David C. Del Rey Fernández & Mark H. Carpenter & Lisandro Dalcin & Stefano Zampini & Matteo Parsani, 2020. "Entropy stable h/p-nonconforming discretization with the summation-by-parts property for the compressible Euler and Navier–Stokes equations," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-54, April.
    2. Zhizhuang Zhang & Xiangyu Zhou & Gang Li & Shouguo Qian & Qiang Niu, 2023. "A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws," Mathematics, MDPI, vol. 11(12), pages 1-18, June.
    3. Gaburro, Elena & Öffner, Philipp & Ricchiuto, Mario & Torlo, Davide, 2023. "High order entropy preserving ADER-DG schemes," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    4. Abgrall, Rémi & Busto, Saray & Dumbser, Michael, 2023. "A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    5. Qiao, Dianliang & Lin, Zhiyang & Guo, Mingmin & Yang, Xiaoxia & Li, Xiaoyang & Zhang, Peng & Zhang, Xiaoning, 2022. "Riemann solvers of a conserved high-order traffic flow model with discontinuous fluxes," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    6. Anandan, Megala & Raghurama Rao, S.V., 2024. "Entropy conserving/stable schemes for a vector-kinetic model of hyperbolic systems," Applied Mathematics and Computation, Elsevier, vol. 465(C).
    7. Li, Gang & Li, Jiaojiao & Qian, Shouguo & Gao, Jinmei, 2021. "A well-balanced ADER discontinuous Galerkin method based on differential transformation procedure for shallow water equations," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    8. Hendrik Ranocha & Manuel Quezada Luna & David I. Ketcheson, 2021. "On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations," Partial Differential Equations and Applications, Springer, vol. 2(6), pages 1-26, December.
    9. Yu, Hao & Wu, Boying & Zhang, Dazhi, 2018. "A generalized Laguerre spectral Petrov–Galerkin method for the time-fractional subdiffusion equation on the semi-infinite domain," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 96-111.

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