# ITERATIVE SOLUTION OF IMPLICIT APPROXIMATIONS OF MULTIDIMENSIONAL PARTIAL DIFFERENTIAL EQUATIONS

**L Herbert**

CiteWeb id: 20160000014

CiteWeb score: 1460

Summary. A new iterative method has been developed for solving the large sets of algebraic equations that arise in the approximate solution of multidimensional partial differential equations by implicit numerical techniques. This method has several advantages over those now in use. First, its rate of convergence does not depend strongly on the nature of the coefficient matrix of the equations to be solved. Second, it is not sensitive to the choice of iteration parameters, and as a result, suitable parameters can be estimated from the coefficient matrix. Finally, it reduces significantly the computational effort needed to solve a set of equations. For a typical set of 961 equations, it was found to reduce the number of calculations by a factor of three, when compared to the most competitive of the older methods. It is expected that this advantage will be even greater for larger sets of equations. 1. Introduction. Approximate solutions of multidimensional differential equations often are obtained by the application of implicit finite difference analogues. A difference equation is written for each grid point in the region of interest, and the resulting set of simultaneous equations must be solved for each time step. Such sets of equations can be solved directly by elimination or by one of several iterative methods, such as relaxation, successive overrelaxation, or ADI (alternating direction iteration). The purpose of this paper is to describe a new iterative procedure that converges much faster than any of these methods. The simplest method of solving these sets of equations is direct solution by elimination. Although this approach is the most efficient method available for small sets of equations, it is not for large sets. The procedure requires 2n2 arithmetic operations to solve n equations of the type being considered. When n becomes relatively large, it is more efficient to use an

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