Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations
Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations
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DOI：
10.1016/j.jcp.2012.07.004
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发表时间：
2012-09
期刊：
影响因子：
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通讯作者：
Wei Shi;Xinyuan Wu;J. Xia
中科院分区：
文献类型：
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作者：
Wei Shi;Xinyuan Wu;J. Xia
本文研究了解在时间和/或空间上具有振荡行为的哈密顿波动方程的积分问题。本文主要研究了多辛扩展Runge-Kutta-Nyström （ERKN）离散化及其离散守恒律。我们首先证明了分别在空间和时间上使用两种辛ERKN方法对哈密顿波动方程进行离散化可以得到显式多辛积分器（Eleap-frogI）。然后利用时间上的辛ERKN法和辛分区龙格-库塔法推导出另一种多辛离散化方法，该方法在空间上等价于众所周知的Störmer-Verlet方法（eleep - frogii）。这两种新的多辛格式是对跳蛙法的扩展。分析了新格式的数值稳定性和色散特性。将两种新的显式多辛方法和跳跃法应用于线性波动方程和正弦-戈登方程，并进行了数值比较实验。数值计算结果证实了这种新型积分器的优越性能和在结构保护方面的显著优势。
In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge–Kutta–Nyström (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERKN methods in space and time respectively lead to an explicit multi-symplectic integrator (Eleap-frogI). Then we derive another multi-symplectic discretization using a symplectic ERKN method in time and a symplectic partitioned Runge–Kutta method, which is equivalent to the well-known Störmer–Verlet method in space (Eleap-frogII). These two new multi-symplectic schemes are extensions of the leap-frog method. The numerical stability and dispersive properties of the new schemes are analyzed. Numerical experiments with comparisons are presented, where the two new explicit multi-symplectic methods and the leap-frog method are applied to the linear wave equation and the Sine–Gordon equation. The numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation.