浅水波方程的高阶保正Well-Balanced ADER间断Galerkin格式
High Order Well-Balanced ADER Discontinuous Galerkin Scheme for Shallow Water Wave Equations
摘要: 本文针对具有不规则几何形状和非平坦底地形的浅水波方程，引入了保正高阶ADER间断Galerkin方法，该方法能准确地保持静水的稳态。为了满足well-balanced的性质，我们提出了well-balanced的数值通量，并基于分解算法将数值解分解为两部分，构造了一种新的源项近似，并相应地将源项近似分解为两部分。此外，还引入了一个简单的保正限制器，从而在干湿锋面附近提供高效和鲁棒性的模拟。大量的数值实验也表明，所得到的格式s能够准确地捕捉静止稳定状态下湖泊的小扰动，保持水面高度的非负性，同时保持光滑解的真正高阶精度。
Abstract: In this paper, we introduce positivity-preserving high order ADER discontinuous Galerkin Methods for the shallow water equations with irregular geometry and a non-flat bottom topography, which maintain the still water steady state exactly. To achieve the well-balanced property, we propose the well-balanced numerical fluxes, and construct a novel source term approximation by decomposing the numerical solutions into two parts based on decomposition algorithm, and resolving the ap-proximation to the source term into two parts accordingly. Moreover, a simple positivity-preserving limiter is introduced in one dimension and then extended to two dimensions to provide efficient and robust simulations near the wetting and drying fronts. Extensive one and two dimensional sim-ulations also indicate that the resulting schemes enjoy the ability to accurately capture small per-turbations to the lake at rest steady state, maintain the non-negativity of the water height, and keep the genuine high order accuracy for smooth solutions at the same time.
文章引用：周翔宇, 张志壮, 钱守国, 李刚. 浅水波方程的高阶保正Well-Balanced ADER间断Galerkin格式[J]. 应用数学进展, 2023, 12(8): 3728-3743. https://doi.org/10.12677/AAM.2023.128367

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