Laplace and Poisson equation solution by relax3d

Abstract

The program RELAX3D is useful for the solution of the three-dimensional, second-order, elliptic partial differential equation, ∇2V(x,y,z)=F(x,y,z), which includes Laplace’s equation and Poisson’s equation. It uses the finite-difference algorithm known as symmetric successive over-relaxation. After a brief survey of other solution techniques, the program is described in detail and several examples of its actual use are given. The program is interactive and accurately solves for V according to the function F and the arbitrary boundary conditions, both supplied via the user-encoded subroutine ‘‘BND.’’ The functions V and F are defined on grids of two or three dimensions, in Cartesian, cylindrical, or polar coordinates. To improve the efficiency of the repetitive five-point or seven-point relaxation iterations on the grid, conventional fortran array addressing is replaced by a fast hard-coded addressing method. In the inner loop where the program spends most of its time, references to the V and F arrays are accomplished using only one addition per address. Detailed derivations are provided for 2D and 3D Cartesian relaxation formulas, with error terms of 𝒪 (h4), 𝒪 (h6), and 𝒪 (h8); we report a set of test computations which verify these formulas. RELAX3D provides alphanumeric printer output with input boundaries clearly displayed, and high-resolution printer or plotter contour maps of any desired slices through the solution region.