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A well-balanced ADER discontinuous Galerkin method based on differential transformation procedure for shallow water equations

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  • Li, Gang
  • Li, Jiaojiao
  • Qian, Shouguo
  • Gao, Jinmei

Abstract

This article develops a new discontinuous Galerkin (DG) method on structured meshes for solving shallow water equations. The method here applies the one-stage ADER (Arbitrary DERivatives in time and space) approach for the temporal discretization and employs the differential transformation procedure to express spatiotemporal expansion coefficients of the solution through low order spatial expansion coefficients recursively. Numerical fluxes using the hydrostatic reconstruction together with a simple source term discretization results in a well-balanced method accordingly. Compared with the Runge-Kutta DG (RKDG) methods, the proposed method needs less computer memory storage due to no intermediate stages. In comparison with traditional ADER schemes, this method avoids solving the generalized Riemann problems at inter-cells. The differential transformation procedure used here is cheaper than the Cauchy-Kowalewski procedure in traditional ADER schemes. In summary, this method is one-step, one-stage, fully-discrete, and easily achieves arbitrary high order accuracy in time and space. Theoretically as well as numerically, the proposed method is verified to be well-balanced. Numerical results illustrate the high order accuracy as well as good resolutions for discontinuous solutions. Moreover, this method is more efficient than the Runge-Kutta DG (RKDG) method.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:395:y:2021:i:c:s0096300320308018
    DOI: 10.1016/j.amc.2020.125848
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    References listed on IDEAS

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    1. 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.
    2. Yuan, Xu-hua, 2018. "A well-balanced element-free Galerkin method for the nonlinear shallow water equations," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 46-53.
    3. Li, Gang & Caleffi, Valerio & Qi, Zhengkun, 2015. "A well-balanced finite difference WENO scheme for shallow water flow model," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1-16.
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    Cited by:

    1. 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.

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