带有几何源项的浅水波方程组的高精度熵稳定有限差分格式
A High Order Entropy Stable Finite Difference Scheme for Shallow Water Equations with Geometric Source Term
DOI: 10.12677/AAM.2023.126301, PDF,    科研立项经费支持
作者: 张志壮, 周翔宇, 高金梅：青岛大学数学与统计学院，山东 青岛
关键词: 浅水波方程组；熵稳定；有限差分格式；高精度；Shallow Water Wave Equations； Entropy Stability； Finite Difference Scheme； High Accuracy
摘要: 本文建立了浅水波方程组的有限差分格式，该格式具有高阶精度和保持熵稳定性。首先，我们构造了一个二阶熵守恒格式，该格式满足给定熵对的熵等式，并能准确地保持湖泊静止稳态。主要思想是使源项的离散化与通量梯度项的离散化相匹配。其次，以具有合理熵守恒通量的二阶熵守恒格式为基础，实现了高阶熵守恒格式。第三，在已有的熵守恒格式上增加适当的耗散项，得到满足离散熵不等式的半离散熵稳定格式。特别是熵稳定格式可以避免熵守恒格式的振荡。其中，耗散项的建立采用根据熵变量构造的加权本质无振荡重构。最后，采用龙格–库塔方法进行时间离散化。文中给出了大量的数值算例，以验证该方法的高精度性及解决捕捉间断的能力。
Abstract: In this article, a finite difference scheme for shallow water wave equations with geometric source terms is established, which has high order accuracy and maintains entropy stability. First, we con-struct a second order entropy conservation scheme that satisfies the entropy conservation equation for a given entropy pair and can accurately maintain the stationary state of the lake. The main idea is to match the discretization of the source term with the discretization of the flux gradient term. Secondly, based on the second order entropy conservation scheme with reasonable entropy con-servation flux, a higher order entropy conservation scheme is implemented. Thirdly, by adding ap-propriate dissipation terms to existing entropy conservation schemes, a semi discrete entropy sta-ble scheme satisfying discrete entropy inequality is obtained. In particular, the entropy stable scheme can avoid the oscillation of the entropy conservation scheme. The dissipation term is estab-lished using weighted essentially non oscillatory reconstruction constructed from entropy variables. Finally, the Runge-Kutta method is used for time discretization. A large number of numerical exam-ples are given to verify the high accuracy of the method and its ability to solve capture discontinui-ties.
文章引用：张志壮, 周翔宇, 高金梅. 带有几何源项的浅水波方程组的高精度熵稳定有限差分格式[J]. 应用数学进展, 2023, 12(6): 2998-3010. https://doi.org/10.12677/AAM.2023.126301

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