非守恒双曲方程组的路径守恒ADER间断Galerkin方法：在浅水方程中的应用
A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Equations: Applications in Shallow Water Equations
DOI: 10.12677/AAM.2023.127337, PDF,  被引量    科研立项经费支持
作者: 赵晓旭, 刘仁迪, 钱守国, 李 刚*：青岛大学数学与统计学院，山东 青岛
关键词: 非守恒双曲方程组；ADER的方法；DG方法；DT过程；Non-Conservative Hyperbolic Equations； ADER Approach； DG Method； DT Procedure
摘要: 本文提出了求解非守恒双曲型偏微分方程的一种新的路径守恒间断Galerkin (DG)方法。特别地，这里的方法采用了一级ADER (在空间和时间的任意导数)方法来实现时间离散化。此外，该方法采用微分变换(DT)过程而不是Cauchy-Kowalewski (C-K)过程来实现局部时间演化。与经典的ADER方法相比，该方法不需要求解内部单元的广义黎曼问题。与RKDG (Runge-Kutta DG)方法相比，该方法不需要中间步骤，因此需要较少的计算机存储空间。简而言之，当前的方法是一步一步完全离散的。而且，该方法在空间和时间上都容易获得高阶精度。浅水方程的数值结果表明，该方法具有较高的阶精度，对间断解具有较好的分辨率。
Abstract: In this article, we propose a new path-conservative discontinuous Galerkin (DG) method to solve the non-conservative hyperbolic partial differential equations (PDE). In particular, the method here applies the one-stage ADER (Arbitrary DERivatives in space and time) approach to fulfill the tem-poral discretization. In addition, this method uses the differential transformation (DT) procedure other than the Cauchy-Kowalewski (C-K) procedure to achieve the local temporal evolution. Com-pared 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 no 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 in space and in time. Numerical results for shallow water equations (SWEs) show that the method enjoys high-order accuracy and keeps good resolutions for discontinuous so-lutions.
文章引用：赵晓旭, 刘仁迪, 钱守国, 李刚. 非守恒双曲方程组的路径守恒ADER间断Galerkin方法：在浅水方程中的应用[J]. 应用数学进展, 2023, 12(7): 3381-3397. https://doi.org/10.12677/AAM.2023.127337

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