Accelerated Solution of Non-Linear Navier-Stokes
         Problems using Chebyshev Iteration Polynomial Based
                        Runge-Kutta Recursions

          A. A. Lorber, G. F. Carey, S. W. Bova, C. H. Harle

               Computational Fluid Dynamics Laboratory
                  The University of Texas at Austin

                               Abstract

The connection between the solution of linear systems of equations by
iterative methods and explicit time stepping techniques is used to
accelerate the solution of ODE systems arising from discretized PDE's
which may involve either physical or artificial transient terms.
Specifically, a class of Runge-Kutta time integration schemes with
extended stability domains has been used to develop recursion formulas
which lead to accelerated iterative performance.  The parameters for
the RK schemes are chosen based on the theory of Chebyshev iteration
polynomials in conjunction with a linear stability analysis.
Recursion performance results are presented for model linear
convection and convection-diffusion problems as well as non-linear
Navier-Stokes fluid flow problems discretized by both
finite-difference and finite-element methods. Of particular interest
is the manner in which the increase in performance is obtained by
straightforward implementations of the new Runge-Kutta schemes.
Application of the schemes and performance results on Cray T90 and
Cray T3D supercomputers will also be presented.