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A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations

Author

Listed:
  • Xiaoxu Zhao

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

  • Baining Wang

    (Qingdao No. 58 High School, Qingdao 266100, China)

  • Gang Li

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

  • Shouguo Qian

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

Abstract

In this article, we propose a new path-conservative discontinuous Galerkin (DG) method to solve non-conservative hyperbolic partial differential equations (PDEs). In particular, the method here applies the one-stage ADER (Arbitrary DERivatives in space and time) approach to fulfill the temporal discretization. In addition, this method uses the differential transformation (DT) procedure rather than the traditional Cauchy–Kowalewski (CK) procedure to achieve the local temporal evolution. Compared with the classical ADER methods, the current method is free of solving generalized Riemann problems at inter-cells. In comparison with the Runge–Kutta DG (RKDG) methods, the proposed method needs less computer storage, thanks to the absence of intermediate stages. In brief, this current method is one-step, one-stage, and fully-discrete. Moreover, this method can easily obtain arbitrary high-order accuracy both in space and in time. Numerical results for one- and two-dimensional shallow water equations (SWEs) show that the method enjoys high-order accuracy and keeps good resolution for discontinuous solutions.

Suggested Citation

  • Xiaoxu Zhao & Baining Wang & Gang Li & Shouguo Qian, 2024. "A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations," Mathematics, MDPI, vol. 12(16), pages 1-21, August.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:16:p:2601-:d:1461997
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    References listed on IDEAS

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    1. 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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