Abstract: 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 KdV equation, and study the scheme based on a Hamilton-space system. Moreover, we prove that the scheme preserves a dispersed multi-symplectic 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.

Key words: Hamilton system, Euler-box scheme, multi-symplectic algorithm, high-order wave equation of KdV type