Testing for effects of asymmetry and instability on preconditioned iterations of conjugate gradient type
R. B. STMPSON · IMA Journal of Numerical Analysis · 1994
We develop a parametrized family of matrices and use them to test the performance of some preconditioned iterative methods as we vary the asymmetry and stability of the test matrices. The test matrices are based on a simple discretization of a dynamic, two-species, contant coefficient, reaction-diffusion system of partial differential equations. The reaction coefficients provide natural parameters for varying the properties of the test matrices, which are typical of modelling applications. 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 parameters than the ‘unreduced’ test matrices. The iterative methods tested are of conjugate gradient type, using incomplete factorization preconditioning. The components of the methods tested are: the acceleration technique (conjugate gradient squared, stabilized biconjugate gradi ent, 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 Out by extensive sampling in regions of the parameter space. The results appear to confirm observations of other studies using diffusion-convection based tests, and, in particular, show that in these instances the performance of the methods is essentially unaffected by asymmetry, but is strongly affected by instability.