ON THE DERIVATION OF HIGHEST-ORDER COMPACT FINITE DIFFERENCE SCHEMES FOR THE ONE- AND TWO-DIMENSIONAL POISSON EQUATION WITH DIRICHLET BOUNDARY CONDITIONS

The primary aim of this paper is to answer the question, What are the highest-order five-or nine-point compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one-and two-dimensional Poisson equation on uniform, quasi-uniform, and nonuniform face-to-face hyperrectangular grids and directly prove the existence or nonexistence of their highest-order local accuracies. Our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights. For the one-dimensional problem, the derivation using the three-point stencil on both "uniform and nonuniform grids yields a scheme with arbitrarily high-order local accuracy. However, for the two-dimensional problem, the derivation using the corresponding five-point stencil on uniform and quasi-uniform grids yields a scheme with at most second-order local accuracy, and on nonuniform grids yields at most first-order local accuracy. When expanding the five-point stencil to the nine-point stencil, the derivation using the nine-point stencil on uniform grids yields at most sixth-order local accuracy, but on quasi-and nonuniform grids yields at most fourth-and third-order local accuracy, respectively.