Multi-symplectic algorithm and simulation of soliton for KdV equation

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 Preissman 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.