KdV方程的多辛算法及其孤子解的数值模拟

Multi-Symplectic Algorithm and Simulation of Soliton for KdV Equation

基于Hamilton空间体系的多辛理论研究了KdV方程。导出了KdV方程的多辛形式及其多种守恒律,并构造了相应的Preissman多辛离散格式及其等价形式。孤子解数值模拟的结果表明:文中构造的多辛格式是有效的,该格式能较好地保持系统的能量和动量特性,并具有良好的长时间数值行为及稳定性。

Aim. Many practical problems are nonlinear. Linearization often brings low accuracy and poor long-time numerical behavior. We now utilize the developing theory of multi-symplecticity to present an algorithm that can bypass linearization. In the full paper, we explain our multi-symplectic algorithm in some detail; in this abstract, we just add some pertinent remarks to listing the two topics of explanation. The first topic is. the multi-symplectic formulation of the KdV equation and its conservation laws. In the first topic, our contribution consists of eqs. （5） through （12） in the full paper; eq. （6） or eq. （7） is the multi-symplectic formulation; eqs. （8）, （10） and （12） are conservation laws. The second topic is： the multi-symplectic Preissman scheme and its equivalent form. The well known Pressman scheme is rewritten as eq. （13） and its equivalent form, eq. （17）, is derived by us. Finally, the results of a numerical experiment for simulating soliton of the KdV equation, given in Figs. 1 and 2 in the full paper, show preliminarily that our multi-symplectic algorithm is good in accuracy and its long-time numerical behavior is also good.