一类发展方程初边值问题差分法的收敛性

Convergence of difference method for initial boundary value problem of a kind of evolution equation

研究用差分法求解自治的发展方程初边值问题时稳定性和收敛性之间的联系.引入反投影算子将发展方程初边值问题的差分格式转化为与初值问题差分格式类似的逐步推进的形式,从而得出:满足Von Neumann条件的差分格式是稳定的格式;在相容条件下,差分格式若稳定(或满足VonNeumann条件)则格式收敛,且对古典解的差分逼近有误差估计式,不再需要线性的条件.

A study was made on the relationship between the stability and convergence when the difference method was used to solve the initial boundary value problem of the autonomous evolution equation. The difference schema of the initial boundary value problem was converted to the step-by-step form, which was similar to the difference schema of the initial value problem. It is concluded that the difference schema satisfying the Von Neumann condition is a stable schema, and that, under the consistency condition, such stable schema is convergent. Moreover, with error estimate expression for the difference approximation to the classical solution, the linearity condition is unnecessary.