Energy conserving local discontinuous Galerkin methods for the improved Boussinesq equation
Energy conserving local discontinuous Galerkin methods for the improved Boussinesq equation
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DOI：
10.1016/j.jcp.2019.109002
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发表时间：
2020-01
期刊：
影响因子：
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通讯作者：
Xiaole Li;Weizhou Sun;Y. Xing;Ching-Shan Chou
中科院分区：
文献类型：
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作者：
Xiaole Li;Weizhou Sun;Y. Xing;Ching-Shan Chou
boussinesq型方程描述了弱非线性长波在浅水中的传播，广泛应用于浅海和港口水波的模拟。本文提出了一种高阶局部不连续Galerkin方法来求解改进的Boussinesq方程，并结合显式跳跃和隐式中点节能时间离散。所提出的全离散方法可以显示为保存连续解的质量和能量的离散版本。给出了半离散方法具有最优收敛阶的误差估计。我们的数值实验证实了最优的收敛速度以及质量和能量的节省性，并表明由于能量的节省性，数值解的误差不会随着时间的推移而显著增加。一系列数值实验表明，该方法能够很好地模拟两孤立波相互作用、单波破碎和爆炸行为。
The Boussinesq-type equations describe the propagation of weakly non-linear long waves in shallow waters and are widely applied to model water waves in shallow seas and harbors. In this paper, we propose a high-order local discontinuous Galerkin method to solve the improved Boussinesq equation, coupled with both explicit leap-frog and implicit midpoint energy-conserving time discretization. The proposed full-discrete method can be shown to conserve the discrete versions of both mass and energy of the continuous solution. The error estimate with optimal order of convergence is provided for the semi-discrete method. Our numerical experiments confirm optimal rates of convergence as well as the mass and energy conserving property, and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property. A series of numerical experiments are provided to show that the proposed method has the capability to simulate the interaction between two solitary waves, single wave break-up and blow-up behavior well.