广义Korteweg-de Vries方程的高精度差分格式

对广义Korteweg-de Vries(generalized Korteweg-de Vries，GKdV)方程的初边值问题进行数值研究，提出一个2层非线性守恒差分格式，该格式的收敛阶为O(τ2+h4)。证明该格式在离散意义下保持原问题质量守恒和能量守恒，分别运用离散能量法和Von Neumann分析法证明该格式的可解性和绝对稳定性。数值实验结果表明，本文格式在时间和空间方向上分别具有2阶和4阶精度，且是质量和能量守恒的。

High-order Finite Difference Scheme for the Generalized Korteweg-de Vries Equation

The initial boundary value problem for the generalized Korteweg-de Vries(GKdV) equation was numerically studied,and a two-level nonlinear conservative difference scheme was proposed,whose convergence order was O(τ2+h4). It was proved that the scheme maintained the mass conservation and energy conservation of the original problem in a discrete sense.The discrete energy method and the Von Neumann analysis method were used to prove the solvability and absolute stability of the scheme.Numerical experimental results showed that the scheme had second and fourth-order accuracy in time and space directions,respectively,and was conserved in mass and energy.