求解广义正则长波方程的新型守恒差分方法

A New Conservative Difference Method for Solving Generalized Regularized Long Wave Equations

对广义正则长波方程的初边值问题提出了两个新的守恒差分格式,即两层线性守恒差分格式和三层非线性守恒差分格式.新格式合理地模拟了广义正则长波方程初边值问题的守恒律,对差分解进行了先验性估计,其中运用数学归纳法推导了解的存在性,并且运用能量分析法对差分格式的稳定性和收敛性进行了分析,数值算例表明新格式的有效性和可行性.由于使用两个守恒差分格式分别求解广义正则长波方程不同时间层,使得能量守恒效果良好.特别地,当选取适当的权系数,计算精度会大幅提高.

Two new conservation difference schemes for an initial boundary value problem of the generalized regularized long wave equation are considered, which are two level linear conservative difference scheme and three level nonlinear conservative difference scheme. The new conservation difference schemes simulate the conservation law of the initial boundary value problem reasonably. It carries on the prior estimation for the difference solution. The present study derives the existence of the solution using mathematical induction, and the stability and convergence of the difference scheme are analyzed by using energy analysis method. Numerical results show that the new conservation difference scheme is feasible and effective. By using two conservative difference schemes to solve the different time levels of the generalized regularized long wave equation respectively, the effect of energy conservation is good. In particular, when the appropriate weight coefficient is selected, the computational accuracy will be improved greatly.