双曲守恒律方程的高精度ADER间断Galerkin方法
High Order ADER Discontinuous Galerkin Method for Hyperbolic Conservation Laws
摘要: 本文提出了一种全新的间断Galerkin (DG)方法，该方法使用单级ADER (任意时–空导数)方式进行时间离散。该方法利用微分变换步骤递归地将解的时–空展开系数通过低阶空间展开系数来表示，能够在空间和时间上达到任意高阶精度。与传统有限体积ADER格式相比较，该方法避免了在单元界面处求解广义Riemann问题。与多级Runge-Kutta DG (RKDG)方法相比较，由于不存在中间级，本方法需要较少的计算机内存。综上所述，所得到的方法是单步的、单级的、全离散的。最后，经典数值算例验证了该方法的良好性能：高精度、高分辨率、高效率。
Abstract: This article develops a new discontinuous Galerkin (DG) method using the one-stage ADER (Arbitrary DERivatives in time and space) approach for the temporal discretization. This current method employs the differential transformation procedure recursively to express the spatiotemporal expansion coefficients of the solution through the low order spatial expansion coefficients, which enables the method to easily achieve arbitrary high order accuracy in space and time. In comparison with the traditional ADER schemes, this method avoids solving the generalized Riemann problems at cell interfaces. Compared with the Runge-Kutta DG (RKDG) methods, the proposed method needs less computer memory storage due to no intermediate stages. In summary, the resulting method is one-step, one-stage, fully-discrete, and easily achieves arbitrary high order accuracy in space and time. Several examples illustrate the good performances of the present method: high order accuracy for smooth solutions, good resolution for discontinuous solutions and high efficiency.
文章引用：张莹娟, 李姣姣, 李刚. 双曲守恒律方程的高精度ADER间断Galerkin方法[J]. 应用数学进展, 2020, 9(8): 1263-1272. https://doi.org/10.12677/AAM.2020.98148

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