Multi-Symplectic Runge—Kutta Collocation Methods for Hamiltonian Wave Equations
Multi-Symplectic Runge—Kutta Collocation Methods for Hamiltonian Wave Equations
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DOI：
10.1006/jcph.1999.6372
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发表时间：
2000-01
影响因子：
4.1
通讯作者：
S. Reich
中科院分区：
文献类型：
--
作者：
S. Reich
一些保守的偏微分方程，像各种波动方程，允许一个多辛公式，可以被看作是一个推广的辛结构的哈密顿常微分方程。我们证明高斯？空间和时间中的勒让德配置导致多辛积分器，即，保持辛守恒律的数值方法类似于辛方法下的辛性守恒，用于Hamilton常微分方程。我们还讨论了能量和动量守恒的问题。从高斯时间离散化开始？Legendre方法计算量较大，本文提出了几种基于Gauss？空间上的勒让德配置和时间上的显式或线性隐式辛离散。
A number of conservative PDEs, like various wave equations, allow for a multi-symplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that Gauss?Legendre collocation in space and time leads to multi-symplectic integrators, i.e., to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for Hamiltonian ODEs. We also discuss the issue of conservation of energy and momentum. Since time discretization by a Gauss?Legendre method is computational rather expensive, we suggest several semi-explicit multi-symplectic methods based on Gauss?Legendre collocation in space and explicit or linearly implicit symplectic discretizations in time.