TESTING FOR EFFECTS OF ASYMMETRY AND INSTABILITY ON PRECONDITIONED ITERATIONS OF CONJUGATE-GRADIENT TYPE

We develop a parametrized family of matrices and use them to test the performance of some preconditioned iterative methods as we vary the as ymmetry and stability of the test matrices. The test matrices are base d on a simple discretization of a dynamic, two-species, contant coeffi cient, reaction-diffusion system of partial differential equations. Th e reaction coefficients provide natural parameters for varying the pro perties of the test matrices, which are typical of modelling applicati ons. These matrices are reducible via a red-black ordering, and it is shown that the reduced matrices are M-matrices for a larger range of p arameters than the 'unreduced' test matrices. The iterative methods te sted are of conjugate gradient type, using incomplete factorization pr econditioning. The components of the methods tested are: the accelerat ion technique (conjugate gradient squared, stabilized biconjugate grad ient, orthomin), the level of fill-in of the incomplete factorization preconditioner, the use of the reduced system, and the effect of time- step size reduction (for dynamic simulations). The tests are carried o ut by extensive sampling in regions of the parameter space. The result s appear to confirm observations of other studies using diffusion-conv ection based tests, and, in particular, show that in these instances t he performance of the methods is essentially unaffected by asymmetry, but is strongly affected by instability.