The Study Of The KdV Equation’s Compact Methods

This paper mainly discusses the compact schemes for several kind of solitary wave equations, including: Kd V equation, Kd V-BBM equation, complex modified Kd V equation and the Kd V-Kd V systems of the Boussinesq equations.Using the six order compact operators of the first order derivative and the second order derivative, we obtain the six order compact difference scheme of the Kd V equation. Based on Fourier stability analysis method to know that the scheme is linearly stable, the convergence orders and conservatives of the scheme are also verified by numerical experiments.The multi-symplectic forms of the Kd V equation, Kd V-BBM equation, complex modified Kd V equation and Kd V-Kd V systems of the Boussinesq equations are derived. Combing midpoint method in time direction with fourth-order compact operator of the first derivative in spatial direction to discrete these multi-symplectic forms, we get the forth-order compact schemes of these equations and proof that these schemes meet the discrete multi-symplectic conservation law. Therefore, these schemes are the forth-order compact multi-symplectic schemes. The convergence orders, long time behavior of solitary wave propagation and conservation laws of these schemes are validated by using the Matlab software, the numerical results show the effectiveness of the proposed algorithms.