High‐order methods for elliptic equations with variable coefficients

Abstract

In this article, we give a simple method for developing finite difference schemes on a uniform square grid. We consider a general, two-dimensional, sezond-order, partial differential equation with variable coefficients. In the case of a nine-point scheme, we obtain the known results of Young and Dauwalder in a fairly elegant fashion. We show how this can be extended to obtain fourth-order schemes on thirteen points. We derive two such schemes which are attractive because they can be adapted quite easily to obtain formulas for grid points near the boundary. In addition to this, these formulas only require nine evaluations for the forcing function. Numerical examples are given to demonstrate the typical performance of one of the fourth-order schemes.

Symbolic derivation of finite difference approximations for the three-dimensional Poisson equation

“…Compared to the code for the 7-point scheme, we add 12 new grid points and 6 constraints, and require the polynomial for u to have a higher degree. Eliminate[{eq [1],eq [2],eq [3],eq [4],eq [5],eq [6],eq [7],eq [8],eq [9], eq [10],eq [11],eq [12],eq [13] This 19-point formula is related to the Mehrstellen scheme for the 2D Poisson equation [8] and has been obtained by various authors [1,6]. We show in Section IV that this scheme is stable and achieves fourth-order accuracy.…”

“…Procedures (based on the truncated Taylor series expansions) for deriving high-order difference approximations for linear partial differential equations have been extensively described in the literature by Gupta, Manohar, Stephenson, and others [1][2][3][4]. These procedures are based upon expressing the solution locally about a given mesh point as a linear combination of the analytic solutions of the differential equation.…”

“…(1) on the coefficients a i,j,k and b i,j,k in the expansion (2). In particular, we get the four set of constraints: …”

Analysis of algebraic systems arising from fourth‐order compact discretizations of convection‐diffusion equations

“…For two-dimensional (2D) problems, HOC schemes at a given mesh point involve the nearest eight neighboring points and different approaches have been used to obtain these nine-point schemes. Procedures based on truncated Taylor series expansions have been described by Ananthakrishnaiah et al [3,4]. Spotz [5] considers the inclusion of approximations to the leading truncation error terms in the central difference scheme to obtain high-order compact difference schemes.…”

On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions

“…For higher-dimensional problems, compact 19-point and 25-point standard CDS of fourth-order local accuracy were presented for the three-and four-dimensional problem, respectively [4,45,50,56]. Similarly, high-order scheme studies have been performed on related problems such as the Poisson equation with Neumann boundary conditions [6], Laplace equation [52,2], Helmholtz equation [2,34,3,25,51,49,35,41,40,7], biharmonic equation [26,27,46], convection-diffusion equations [8,23,24,32,31,29,30,28,48,47,22,19] with various boundary conditions, and stream function vorticity equations [42,43,44,57].…”