Superconvergence of Solution Derivatives for the Shortley–Weller Difference Approximation for Parabolic Problems

Abstract

In [8-11] a superconvergence analysis is derived for the smooth and singular Poisson equations by the finite difference method (FDM) using the Shortley–Weller approximation. In this article, we explore the superconvergence analysis for a parabolic equation using the Shortley–Weller approximation and the Crank–Nicolson scheme (CNS) in space and time discretization, respectively, denoted simply as FDM-CNS. The results of derivative superconvergence in [8-11] can be extended to parabolic problems of smooth and singular solutions. The main results are as follows: when and , the superconvergence rate O(h 2 + k 2) is derived for the solution derivatives in discrete H 1 norms by the FDM-CNS on rectangular domains, where k is the time mesh spacing in the Crank–Nicolson scheme and h is the maximal mesh length of difference grids used. Note that the difference grids are not confined to be quasi-uniform, and local refinements are allowed for the solutions with unbounded derivatives. Numerical experiments are provided to support the superconvergence O(h 2 + k 2).