High-order implicit finite difference schemes for the two-dimensional Poisson equation

In this paper, a new family of high-order finite difference schemes is proposed to solve the two-dimensional Poisson equation by implicit finite difference formulas of (2M+1) operator points. The implicit formulation is obtained from Taylor series expansion and wave plane theory analysis, and it is constructed from a few modifications to the standard finite difference schemes. The approximations achieve (2M+4)-order accuracy for the inner grid points and up to eighth-order accuracy for the boundary grid points. Using a Successive Over-Relaxation method, the high-order implicit schemes have faster convergence as M is increased, compensating the additional computation of more operator points. Thus, the proposed solver results in an attractive method, easy to implement, with higher order accuracy but nearly the same computation cost as those of explicit or compact formulation. In addition, particular case M=1 yields a new compact finite difference schemes of sixth-order of accuracy. Numerical experiments are presented to verify the feasibility of the proposed method and the high accuracy of these difference schemes.