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

Abstract. 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 non-uniform face-to-face hyper-rectangular grids and directly prove the existence or non-existence 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 non-uniform 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 non-uniform 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 non-uniform grids yields at most fourth- and third-order local accuracy, respectively

Dirac Assisted Tree Method for 1D Heterogeneous Helmholtz Equations with Arbitrary Variable Wave Numbers

In this paper we introduce a method called Dirac Assisted Tree (DAT), which can handle heterogeneous Helmholtz equations with arbitrarily large variable wave numbers. DAT breaks an original global problem into many parallel tree-structured small local problems, which can be effectively solved. All such local solutions are then linked together to form a global solution by solving small Dirac assisted linking problems with an inherent tree structure. DAT is embedded with the following attractive features: domain decomposition for reducing the problem size, tree structure and tridiagonal matrices for computational efficiency, and adaptivity for further improved performance. In order to solve the local problems in DAT, we shall propose a compact finite difference scheme with arbitrarily high accuracy order and low numerical dispersion for piecewise smooth coefficients and variable wave numbers. Such schemes are particularly appealing for DAT, because the local problems and their fluxes in DAT can be computed with high accuracy. With the aid of such high-order compact finite difference schemes, DAT can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems - 4 by 4 matrices in the extreme case - with tridiagonal coefficient matrices in a parallel fashion. Several examples will be provided to illustrate the effectiveness of DAT and compact finite difference schemes in numerically solving heterogeneous Helmholtz equations with variable wave numbers. We shall also discuss how to solve some special two-dimensional Helmholtz equations using DAT developed for one-dimensional problems. As demonstrated in all our numerical experiments, the convergence rates of DAT measured in relative $L_2$, $L_\infty$ and $H^1$ energy norms as a result of using our M-th order compact finite difference scheme are of order M