Iterative solution of the nonlinear parabolic periodic boundary value problem

In this paper, the successive over-relaxation method (S.O.R.) is outlined for the numerical solution of the implicit finite difference equations derived from the Crank-Nicolson approximation to a mildly non-linear parabolic partial differential equation with periodic spatial boundary conditions. The usual serial ordering of the equations is shown to be inconsistent, thus invalidating the well known S.O.R. theory of Young (1954), but a functional relationship between the eigenvalues of the S.O.R. operator and the Jacobi operator of a closely related matrix is derived, from which the optimum over-relaxation factor, wb, can be determined directly. Numerical experiments confirming the theory developed are given for the chosen problem.