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

Xiaoxu Zhao, Baining Wang, Gang Li () and Shouguo Qian
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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

Mathematics, 2024, vol. 12, issue 16, 1-21

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.

Keywords: non-conservative hyperbolic systems; ADER approach; DG method; DT procedure (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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