Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations

Abstract

In the solution by numerical methods of ordinary and partial differential equations, the derivatives are replaced by their finite-difference equivalents, usually infinite series of differences. For the employment of relaxation methods it has been customary to ignore all but the dominant first term of these series, taking sufficient intervals to ensure the effective vanishing of the neglected terms. In this paper, methods are developed whereby the full difference equations can be used at the maximum interval consistent with the convergence of the differences. The number of mesh points, and hence the labour and difficulty of the relaxation process, is thus considerably reduced, and increased accuracy is obtained. The method is applied to eight examples, including ordinary and partial differential equations, eigen-value problems, and the more difficult problem of curved boundaries.