An asymmetrical finite difference network

Abstract

Introduction. Finite difference techniques have been used extensively in recent years in the solution of two-dimensional second order boundary value problems that have proved to be intractable by other methods. The differential equation is replaced by a system of linear algebraic equations, the solution of which gives the values of the wanted function at a finite number of points lying at the intersections of a gridwork. The use of regular polygons, either squares or equilateral triangles, in the formation of these grid-works has the desirable property that the equations associated with each node (inter-section) point have a particularly simple, symmetrical form that is identical for all interior points. There are, however, two troublesome problems which arise in connection with the use of regular polygons. The first of these arises when the region has curved boundaries. In such cases some node points near the boundary will be connected to the boundary by gridwork elements of irregular lengths, necessitating the use of special equations for these points. The second problem concerns the change of mesh size at points within the boundary. It is frequently uneconomical from the point of view of the labor of computation to use the same mesh size at all points. In the neighborhood of a sharp corner or near other types of singularities, the mesh size must be reduced if an accuracy is to be obtained that is comparable with the accuracy of the solution in parts of the region where the behaviour of the wanted function is more uniform. Both of these problems have received attention from writers on relaxation methods and it is with these problems that the present paper is principally concerned. A method will be described by means of which the coefficients of the system of algebraic equations can be computed for an arbitrary distribution of node points. The positions of these node points can then be chosen to fit the boundary conditions and other special requirements of each problem. In the construction of a finite difference gridwork to be used in the solution of physical problems, it is helpful to associate physical properties with the elements of the gridwork. Southwell and his co-workers have regarded the gridwork as a network of tensioned strings [1] while others have regarded the gridwork as a network of electrical elements [2]. In the case of a second order boundary value problem, this clear physical picture of the gridwork is lost if the differential equation is replaced by difference equations involving difference operators higher than the second order. In order to preserve the physical picture and to simplify the calculations, the higher order difference terms are usually regarded as corrections which are added in the final stage of calculation, if the relaxation method is to be used [3]. Another important reason for eliminating higher order difference operators arises in connection with analog computing devices, in which the physical picture of the network is realized. Electrical circuits have been used extensively in the solution of many kinds of boundary value problems [4,5,6]. In the construction of these circuits one consideration enters that is not present when the finite-difference equations are solved by purely numerical methods, namely that the circuit must be physically realizable. If the