On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: Energy conservation and multi-symplecticity
On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: Energy conservation and multi-symplecticity
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DOI：
10.1016/j.jcp.2020.109662
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发表时间：
2019-12
期刊：
影响因子：
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通讯作者：
Zheng Sun;Y. Xing
中科院分区：
文献类型：
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作者：
Zheng Sun;Y. Xing
本文提出并研究了一维多辛Hamilton偏微分方程的间断Galerkin（DG）方法。我们特别专注于半离散格式与空间离散，并表明，建议DG方法可以同时保持多辛结构和能量守恒与一般类的数值通量，其中包括著名的中心和交替通量。讨论了它在波动方程、Benjamin-Bona-Mahony方程、Camassa-Holm方程、Korteweg-de弗里斯方程和非线性薛定谔方程中的应用。一些数值结果证明了所提出的方法的精度和长时间的行为。数值上，我们观察到，某些选择的数值通量在讨论类可能有助于实现更好的精度相比，常用的，包括中心通量。
In this paper, we present and study discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian partial differential equations. We particularly focus on semi-discrete schemes with spatial discretization only, and show that the proposed DG methods can simultaneously preserve the multi-symplectic structure and energy conservation with a general class of numerical fluxes, which includes the well-known central and alternating fluxes. Applications to the wave equation, the Benjamin–Bona–Mahony equation, the Camassa–Holm equation, the Korteweg–de Vries equation and the nonlinear Schrödinger equation are discussed. Some numerical results are provided to demonstrate the accuracy and long time behavior of the proposed methods. Numerically, we observe that certain choices of numerical fluxes in the discussed class may help achieve better accuracy compared with the commonly used ones including the central fluxes.