Hybrid Algorithms For Cyclically Reduced Convection-Diffusion Problems
by
Muddun Bhuruth
Department of Mathematics, University of Mauritius, Reduit, Mauritius
Coauthors: M. Kumar Jain (University of Mauritius), Ravi Boojhawon (University of Mauritius)

We consider hybrid and adaptive iterative algorithms for cyclically-reduced discrete convection-diffusion problems. The motivation behind hybrid methods is that an iterative method such as GMRES is too expensive to run to completion and that restarted versions of the algorithm often result in slow convergence. Various hybrid algorithms have been proposed for the iterative solution of nonsymmetric linear systems. These hybrid methods combine via a two phase algorithm, iterative methods that require no a priori information about the coefficient matrix in the first phase with Chebyshev or Richardson iteration in the second phase.

Our numerical experiments show that, for constant coefficient problems, an adaptive Chebyshev algorithm that uses modified moments to approximate the eigenvalues of the coefficient matrix for the cyclically reduced linear system converge faster than the hybrid algorithms based on GMRES/Richardson methods. For variable coefficient problems, a hybrid GMRES algorithm that eliminates the eigenvalue computation phase and instead uses the GMRES polynomial constructed during the first phase itself as the residual polynomial for second phase has comparable convergence rates to the Chebyshev method based on modified moments. The effect of different orderings on the convergence of the hybrid methods is also studied.

Date received: July 16, 1999