Tensor Product Analysis of Partial Difference Equations

1. Introduction. This note presents a few of the results which we have obtained by applying a classical and fundamental idea to the analysis of certain partial difference equations. The idea is that certain multidimensional problems can be solved by solving a few one-dimensional problems—it is the basis of the classical method of separation of variables of mathematical physics. In the case of partial difference equations, this idea leads to tensor product analysis of the matrices involved. With this approach we accomplish the following: (i) Explicit exact solutions of problems consisting of separable partial difference equations and boundary conditions are obtained, (ii) A stable algorithm is devised with which these exact solutions can be evaluated with less work than approximate solutions can be computed by overrelaxation techniques, (iii) A simple, direct analysis of certain alternating direction implicit (ADI) methods is carried out and, as a result, a simple explanation of the power of this method is given, (iv) A necessary and sufficient condition is found for commutativity of certain matrices which occur in ADI schemes.