Numerical solution of two-point boundary value problems

Abstract

This paper presents an original formulation of two-point boundary value and eigenvalue problems expressed as a system of first-order equations. The fundamental difference between the new method and other methods based on a first-order approach is the introduction of conditions of an integral character to supplement the simultaneous set of first-order equations, which are hence never regarded as an initial value problem. The consideration of integral conditions leads to establish a class of linear multipoint schemes for the numerical solution of boundary value problems for ordinary differential equations. Furthermore, the global character of the integral conditions (nonlocality) combined with the block structure of the system of algebraic equations allow dealing with stiff problems by means of the classical procedure of iterative refinement introduced by Wilkinson. The properties of the numerical schemes are illustrated by the solution of linear and nonlinear problems and by the accurate and efficient determination of some eigensolutions of a difficult problem of hydrodynamic stability. The proposed method is conceptually simpler and numerically more convenient than existing initial value methods, while still retaining all the advantages of a formulation based on a first-order system.