Computation of Natural Convection in a Rectangular Enclosure,"

A direct solution method was used, for the solution of the set of difference equations, requiring only the storage of those ele-ments contained between the two outermost nonzero diagonals of the coefficient matrix. The method used involves the tri-angular decomposition of the coefficient matrix followed by the solution of two triangular systems. Hounding errors present in t he first solution are then reduced to negligible proportions by an iterative method. The theory is as follows. Each set of equations may be written in the form AX = B (45) where A = coefficient matrix B = constant vector X = solution vector Provided A is nonsingular, it may be factorized in the form A = III (40) where L = lower triangular matrix U = upper triangular matrix with unity elements on main diagonal In the case where the nonzero elements of A are all contained within a band around the main diagonal, it may be shown (hat the nonzero elements of L and (J all lie within corresponding-bands. Substituting equation (45) into (46), we obtain