Dichotomy and conjugate gradients in the stiff initial value problem

Abstract

We propose, analyze, and experiment with solution techniques which employ the conjugate gradient algorithm coupled with prediction steps for solving the algebraic equations arising at each mesh point in the numerical development of solutions of the model stiff system x = Ax. A stability and error analysis based on a dichotomization of the solutions of the system into rapidly and slowly decaying modes is made, to demonstrate the numerical stability of these methods. Stiff problems are characterized by this dichotomy, and we note that the conjugate gradient algorithm improves in effectiveness with the exaggeration of this characterization. © 1981.