Analysis of algebraic systems arising from fourth-order compact discretizations of convection-diffusion equations

We study the properties of coefficient matrices arising from high-order compact discretizations of convection-diffusion problems. Asymptotic convergence factors of the convex hull of the spectrum and the field of values of the coefficient matrix for a one-dimensional problem are derived, and the convergence factor of the convex hull of the spectrum is shown to be inadequate for predicting the convergence rate of GMRES. For a two-dimensional constant-coefficient problem, we derive the eigenvalues of the nine-point matrix, and we show that the matrix is positive definite for all values of the cell-Reynolds number. Using a recent technique for deriving analytic expressions for discrete solutions produced by the fourth-order scheme, we show by analyzing the terms in the discrete solutions that they are oscillation-free for all values of the cell Reynolds number, Our theoretical results support observations made through numerical experiments by other researchers on the non-oscillatory nature of the discrete solution produced by fourth-order compact approximations to the convection-diffusion equation. (C) 2002 Wiley Periodicals, Inc.