The error committed by stopping the Newton iteration in the numerical solution of stiff initial value problems

Abstract

This paper concerns the numerical solution of initial value problems, for non-linear ordinary differential equations, by step-by-step methods that are implicit. For non-stiff problems, fixed-point iteration, also called Picard iteration, is a classical process for solving the system of (algebraic) equations occurring in each time step of such step-by-step methods. The order of the error committed by stopping this process, after a fixed number of iteration steps, is well understood. For stiff problems, Picard iteration is not a useful process, so that one almost invariably employs Newton's method or a variant thereof. This paper analyzes the stopping error for such Newton-type iterations. We aim at an understanding of this error that is comparable to the insight one has for the case of Picard iteration. The error committed by stopping a Newton-type iteration, after a specified number of iteration steps, is estimated and related to the pure discretization error of the underlying step-by-step method. Also the question of existence and uniqueness of solutions to the system of (algebraic) equations is studied. All conclusions in the paper are relevant to non-linear problems that may be arbitrarily stiff. The estimates obtained for these problems reflect order reduction phenomena, in the presence of stiffness, confirmed by numerical experiments. The focus is on linear multistep methods and Runge-Kutta methods, but part of the conclusions are also relevant to other situations.