abcd Boussinesq系统的高阶中心间断Galerkin-有限元方法
High Order Central Discontinuous Galerkin-Finite Element Methods for the abcd Boussinesq System
摘要: Boussinesq类型的方程是一类重要的非线性色散浅水波方程，广泛应用于研究小振幅浅水波的传播现象。我们考虑一个含有四个参数的Boussinesq系统：abcd Boussinesq系统，并针对该系统设计了一个高阶中心间断Galerkin-有限元方法。该方法首先将abcd Boussinesq系统改写为守恒律方程与椭圆型方程的一个耦合系统，然后采用中心局部间断Galerkin方法求解守恒律方程，使用有限元方法求解椭圆型方程，最后我们通过一系列的数值算例验证所提出方法的高阶精度和有效性。
Abstract: Boussinesq type equations are an important class of nonlinear dispersive shallow water wave equa-tions, it has been widely studied to model the propagation phenomena of small amplitude shallow water waves. In this work, we consider a Boussinesq system with four parameters: abcd Boussinesq system, and design a high order central discontinuous Galerkin-finite element methods for solving this system. In our numerical approach, we first reformulate the abcd Boussinesq system into a coupled system of conservation law and elliptic equations. Then we propose a family of high order numerical methods which discretize the conservation law with central local discontinuous Galerkin methods and the elliptic equations with continuous finite element methods. Different types of nu-merical tests are provided to illustrate the accuracy and effectiveness of the proposed schemes.
文章引用：夏有伟, 程用平. abcd Boussinesq系统的高阶中心间断Galerkin-有限元方法[J]. 应用数学进展, 2023, 12(10): 4288-4299. https://doi.org/10.12677/AAM.2023.1210422

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