Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations

Abstract

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.