A Fast Implicit Numerical Method for Time Dependent Viscous Flows

The iterative method used by Esch (1964) and Pearson (1964, 1965a, b), for the solution of an implicit finite difference approximation to the Navier Stokes equation, is analysed. A more general iteration method is suggested that may require many iteration parameters, and it is shown how these parameters can be computed. It was found that when the non-dimensional number vΔt/2L² is small, a single optimum iteration parameter exists (v being the kinematic viscosity, Δt the time step and L a characteristic length). An approximate expression for the "best" parameter is developed, and a procedure is described for improving that estimate. With the improved estimate and extrapolation in time, convergence is achieved in one or two iterations per time step on the average. In some cases the time step used was 200 times bigger than the time step required for stability of explicit schemes.