Analysis of a block red-black preconditioner applied to the Hermite collocation discretization of a model parabolic equation (Q2762698)

The numerical solution of a model parabolic partial differential equation in two spatial dimensions discretized by Hermite collocation is investigated. The resulting linear algebraic system is solved via Bi-CGSTAB method of \textit{H. A. van der Vorst} [SIAM J. Sci. Stat. Comput. 13, No. 2, 631-694 (1992; Zbl 0761.65023)] with block Red-Black Gauss-Seidel (RBGS) preconditioner. An analytic formulae for the eigenvalues that control the rate of convergence of this method are presented. These formulae depend on the location of the collocation points and can be utilized to determine where the collocation points should be placed in order to make the Bi-CGSTAB/RBGS method converge as quickly as possible. Strategies for selecting the time-step size are discussed. A complete stability analysis is also included.