Numerical Solution of Elliptic Difference Equations by Matrix Methods

A non-iterative method for solving boundary value problems of elliptic difference equations is developed that is based on the simple structure of the triangular matrices obtained by applying the well-known elimination procedure of Gauss to the matrix of the difference equation. It is proved that the procedure is numerically stable, i. e., there is no tendency to error-growth. The method seems to be particularly suited to such problems as weather forecasting on high-speed computing machines where one has to solve the same equations (e. g., Poisson’s or Helmholtz’s) a great number of times with the same net but different right member and boundary values. The number of stored constants as well as the computational work in the application of the method is only a fractional part of that required when using the inverse matrix (Green’s function). Poisson’s equation for a rectangular region is given a close study, but the method can be applied to irregular regions and to equations of higher order.