A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation

This paper introduces a novel method for deducing high-order compact difference schemes for the two-dimensional (2D) Poisson equation. Like finite volume method, a dual partition is introduced. Combining Simpson integral formula and parabolic interpolation, a family of fourth-order and sixth-order compact difference schemes are obtained based on three different types of dual partitions. Moreover, several new fourth-order compact schemes are gained and numerical experiments are shown two of them are much better than almost any other fourth-order schemes which have been presented in others’ work. The outline for the nonlinear Poisson equation is also given. Numerical experiments are presented to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference schemes.