A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation  

Korteweg-de Vries方程初边值问题的一个二阶收敛线性化差分格式

作　　者：Wang Xuping Sun Zhizhong 王旭平;孙志忠(东南大学数学学院,南京211189)

机构地区：[1]School of Mathematics, Southeast University, Nanjing 210096, China

出　　处：《Journal of Southeast University(English Edition)》2022年第2期203-212,共10页东南大学学报（英文版）

基　　金：The National Natural Science Foundation of China(No.11671081).

摘　　要：To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is constructed for the system.The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable.The energy method is applied to the theoretical analysis of the difference scheme.Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem.Moreover,the difference scheme converges when the step ratio satisfies a constraint condition,and the temporal and spatial convergence orders are both two.Numerical examples verify the convergence order and the invariant of the difference scheme.Furthermore,the step ratio constraint is unnecessary for the convergence of the difference scheme.Compared with a known two-level nonlinear difference scheme,the proposed difference scheme has more advantages in numerical calculation.为求解Korteweg-de Vries方程的初边值问题,首先利用降阶法得到一个等价的耦合非线性方程组,再对该方程组建立差分格式.引进的新变量可以从差分格式中分离,得到仅含有原变量的差分格式,该差分格式在实际计算中,每一时间层上只需要解一个四对角的线性方程组,计算量和存储量都很小.应用能量法对差分格式进行了理论分析,证明了差分格式是唯一可解的,且满足一个与原问题相应的能量守恒律.在步长比满足一个限制条件下,差分格式是收敛的,时间收敛阶和空间收敛阶都为2.数值算例验证了差分格式的收敛阶和数值解满足能量守恒律,且步长比的限制性条件对差分格式的收敛性不是必要的.通过与一个已知的两层非线性差分格式进行对比,所提出的差分格式在数值计算方面更有优势.

关 键 词：Korteweg-de Vries(KdV)equation linearized difference scheme conservation convergence 

分 类 号：O17[理学—数学]