广义变系数KdV方程的保角能量守恒方法

Conformal Energy-Preserving Method for Generalized Variable Coefficient KdV Equation

基于保角哈密尔顿系统的辛形式,对带依时系数的广义KdV(TDKdV)方程提出一个保角能量守恒算法.通过算子分裂方法,方程被分裂成一个哈密尔顿系统和一个耗散系统,其中,耗散系统被精确求解.哈密尔顿系统在时间上采用二阶平均向量场(AVF)方法离散,在空间上采用傅里叶拟谱方法离散.在合适的边界条件下,所提方法可精确保持离散保角能量守恒律及离散保角质量守恒律.数值实验验证文中方法在长时间数值模拟过程中的有效性.

Based on the symplectic formulation of the conformal Hamiltonian system,a conformal energy-preserving algorithm for the generalized KdV equation with time-dependent coefficients(TDKdV)is proposed.The equation is split into a Hamiltonian system and a dissipative system by the operator splitting method in which the dissipative system can be solved exactly.The Hamiltonian system is discretized by the second order average vector field(AVF)method in time and the Fourier pseudo-spectral method in space,and the proposed method can exactly preserve the discrete conformal energy conservation law and the discrete conformal mass conservation law under the appropriate boundary conditions.Numerical experiments verify the effectiveness of the method during the long-time numerical simulations.