一类非线性发展方程差分法的稳定性

研究了用差分法求解自治的发展方程时稳定性和收敛性这两个基本概念之间的联系,利用计算时间的有限性和紧致性,在可解集为开集的条件下,得出方程解的邻近也可解的结论.当近似方法同时具备收敛性和稳定性时,方程解必然具备逐点Lipschitz条件.方程解的邻近如果可解并具备逐点Lipschitz条件,则差分法收敛必有稳定界存在,从而差分格式收敛性保证其稳定性,因此可以放弃线性这一重要条件.

Stability study of difference method for solution of a kind of nonlinear evolution equations

The relationship between the stability and convergence of the solution in theory was studied when the difference method was applied to solving the autonomous evolution equation.Based on the limitation and tightness of computational time,it is concluded that the neighborhood of the equation solution is solvable under the condition of open solvable set.When the approximate method is of the characteristics of convergence and stability,the solution to the equation is certain to meet the point-by-point Lipschitz condition.If the neighborhood of equation solution is solvable and meets the point-by-point Lipschitz condition, stable boundary could be obtained as the convergence of the difference method is realized,and thus the stability is ensured.Therefore,the linearity as an important condition could be omitted.