一类高阶KdV类型水波方程的多辛Euler-box格式

Multi-symplectic Euler-box Scheme for a High-order Wave Equation of KdV Type

高阶KdV类型水波方程作为一类重要的非线性方程有着许多广泛的应用前景.本文主要研究高阶KdV类型水波方程的多辛Euler-box格式.首先,通过正则变换,构造了高阶KdV方程的多辛结构,并得到该系统的多辛守恒律、局部能量守恒律和动量守恒律.然后,我们利用Euler-box格式对高阶KdV方程进行离散,并基于Hamilton空间体系的多辛理论研究了该系统的离散Euler-box格式.我们证明该格式满足离散多辛守恒律,并且给出该格式的向后误差分析.最后,数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.

The high order KdV equation, an important nonlinear wave equation, has a broad application prospect. In the paper, a multi-symplectic Euler-box scheme is presented for the high order KdV equation. First, we give the multi-symplectic structure of the high-order KdV equation by canonical transformation, and obtain an associated multi-symplectic conservation law, the local energy and momentum conservation laws. Then, we apply the Euler-box scheme to obtain a discrete scheme of the high order Kd V equation, and study the scheme based on a Hamilton-space system. Moreover, we prove that the scheme preserves a dispersed multisymplectic conservation law, and give the backward error analysis of the scheme. Finally, the numerical experiments of the solitary wave are given, and results show that the numerical scheme is an efficient method with excellent long-time numerical behaviors.