KdV方程的非标准有限差分格式

Nonstandard Finite Difference Schemes for a KdV Equation

给出了一类KdV方程的精确差分格式和非标准有限差分格式.先构造KdV方程的精确有限差分格式,并由此推导出一个非标准有限差分格式.在构造差分格式中,重点给出步长函数(分母函数)的具体形式,同时证明了该方法可以保持KdV方程解的正性和有界性.通过数值实验验证了非标准有限差分格式的可行性和有效性.

In this paper,we give an accurate difference scheme and nonstandard finite difference scheme for a class of KdV equation.The exact finite difference scheme of the KdV equation is constructed and a nonstandard finite difference scheme is derived.In the construction of the difference scheme,the concrete form of the step function(denominator function) is given,and it is proved that thise method keeps the positivity and boundedness of the solution of the KdV equation.The feasibility and validity of the nonstandard finite difference scheme are verified by numerical experiments.