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

摘要

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,y-space in which the desired solution is contained (here x is the independent and the vector-valued function 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.