A Unified View of Some Methods for Stiff Two-Point Boundary Value Problems

Abstract

The paper analyzes a number of known methods for the solution of two-point boundary value problems for ordinary differential equations; the methods considered have in common that they lead to a sequence of initial value problems. A routine approach fails if the differential equations are stiff. (The meaning of the word stiff applied to two-point boundary value problems is explained in the text.) In geometric terms, all methods perform the same basic task ; they determine a hypersurface in an x, ${\bf y}$-space in which the desired solution is contained (here x is the independent and the vector-valued function ${\bf y}$ the dependent variable). Even in stiff equations, this hypersurface is well-defined provided that the problem is well-posed. Nevertheless, classical methods encounter difficulties; the vectors used to describe the hypersurface have a tendency to become more and more parallel unless one takes special precautions. The method of Godunov–Conte and the parallel shooting method of Kelley periodically change the vectors used in this description. In the interpretation of this article, the method of invariant embedding first derives a partial differential equation for this hypersurface. For linear problems, it is then reduced to a number of initial value problems. The projection method of Guderley and Nikolai leads to the same procedure as invariant embedding; it describes the hypersurface S by projections on suitable subspaces. All of these methods are able to control accuracy losses, but some require fairly small integration steps. Special methods for the solution of stiff initial value problems (for instance, that of Gear) are useful only if the functions which are computed are relatively smooth. They are therefore ineffective if, in intermediate steps of the computation, particular solutions of the original differential equation must be generated. The method of invariant embedding and the projection method is not affected by this restriction. The idea of finding a surface to which the solution is confined can also be carried over with slight modification to two-point boundary value problems for difference equations. One obtains a surprisingly simple rule for the sequence in which the unknowns must be eliminated.