On the Numerical Integration of Nonlinear Two-Point Boundary Value Problems Using Iterated Deferred Corrections. Part 2: The Development and Analysis of Highly Stable Deferred Correction Formulae

Abstract

Iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two-point boundary value problems in ordinary differential equations. However, there may be loss of stability when this procedure is applied, since the process of deferred correction is normally explicit. This phenomenon is well known in the theory of initial value problems, where a classic stability theorem of Dahlquist is central in the design of A-stable methods. As a direct result of the deterioration in stability, deferred correction procedures are sometimes ineffective for problems with “rough” solutions, such as singular perturbation or turning point problems. (Such problems are often referred to as “stiff.”) In this paper we consider the idea of using implicit deferred corrections, which allows the derivation of high-order methods that maintain the good stability properties of the underlying low-order formula. Using the ideas developed in this paper, a highly stable, variable-order deferred correction scheme is derived, and the performance of its implementation is compared with that of COLSYS and D02GAF on some linear test problems.