Local Discontinuous Galerkin Methods for the Boussinesq Coupled BBM System
Local Discontinuous Galerkin Methods for the Boussinesq Coupled BBM System
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DOI：
10.1007/s10915-017-0546-0
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发表时间：
2017-09
影响因子：
2.5
通讯作者：
Joshua Buli;Y. Xing
中科院分区：
文献类型：
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作者：
Joshua Buli;Y. Xing
非线性色散波动方程模拟了大量允许特殊解的物理系统，如孤子和孤波。由于非线性和色散效应的复杂性，高阶数值方法在计算中捕捉物理系统是有效的。本文考虑Boussinesq耦合BBM系统，提出了求解BBM系统的局部间断Galerkin(LDG)方法。对于所提出的LDG方法，我们提供了两种不同的数值通量选择，即迎风通量和交变通量，并建立了它们的稳定性分析。对于具有交变磁通的LDG方法，对线性化的BBM系统进行了误差估计。为了给出一种在数值上守恒哈密顿量的时间离散化方法，提出了离散化中含有非平凡非线性项的中点规则。实现了哈密顿守恒和耗散时间离散，并对数值通量和时间离散的多种组合进行了数值试验。数值算例验证了所提出的耦合BBM系统的LDG方法的精度、长期模拟和哈密顿守恒性。
Nonlinear dispersive wave equations model a substantial number of physical systems that admit special solutions such as solitons and solitary waves. Due to the complex nature of the nonlinearity and dispersive effects, high order numerical methods are effective in capturing the physical system in computation. In this paper, we consider the Boussinesq coupled BBM system, and propose local discontinuous Galerkin (LDG) methods for solving the BBM system. For the proposed LDG methods, we provide two different choices of numerical fluxes, namely the upwind and alternating fluxes, as well as establish their stability analysis. The error estimate for the linearized BBM system is carried out for the LDG methods with the alternating flux. To present a time discretization that conserves the Hamiltonian numerically, the midpoint rule with a nontrivial nonlinear term in the discretization is proposed. Both Hamiltonian conserving and dissipating time discretizations are implemented, with multiple combinations of numerical flux and time discretization tested numerically. Numerical examples are provided to demonstrate the accuracy, long-time simulation, and Hamiltonian conservation properties of the proposed LDG methods for the coupled BBM system.