Mathematical and Computational Aspects of Iterative Chebyshev Schemes for Time Integration of Parabolic Equations.

This talk examines the theoretical and computational foundations of the explicitly iterative Chebyshev LI-M scheme for time integration of multidimensional linear parabolic equations. The design of the Chebyshev LI-M scheme as a method for integrating a stiff system of differential equations arising from the spatial discretization of an initial-boundary value parabolic problem is dictated solely by the requirements of approximation and stability. This fundamentally distinguishes the LI-M scheme from Chebyshev's acceleration of convergence of iterative processes used to solve elliptic equations or implicit schemes for parabolic equations. The necessary limit on the number of iterations to ensure stability is provided by the Gelfand-Lokutsievskii theorem. The LI-M scheme is based on the optimal properties of Chebyshev polynomials. The degree of the polynomial, which determines the number of iterations, is found without using tuning parameters; only an estimate of the maximum eigenvalue of the discrete operator is required, which is easily calculated using Gershgorin's theorem on spectral circles. Issues of scheme accuracy and monotonicity preservation are discussed, and examples of the scheme's use in solving nonlinear parabolic equations in applications are given.