This paper is concerned mainly with the linear differential equation system \(y'(x)=A(x)y(x)+f(x),\) subject to linear two-point boundary conditions \(B_ 0y(0)+B_ 1y(c)=g,\) although non-linear generalizations are also considered. For the linear problem, fundamental difficulties in numerical approximations can arise if A possesses large eigenvalues with real parts of either sign or if eigenvalues of A can change magnitude with great rapidity. The class of problems which exhibit these behaviours are referred to as stiff by analogy with related initial value problems.
By treating scalar problems in preliminary sections, the groundwork is laid for considering high dimensional problems for which the stiffness is confined to certain components which possess a special type of diagonal dominance, in which not only are the sum of the magnitudes of the off- diagonals in a row of A bounded in terms of the magnitudes of the real parts of the corresponding diagonals, but the arguments of the diagonal elements are bounded away from \(\pm \pi /2\). This last requirement is to avoid the additional difficulties associated with highly oscillatory solution components.
For problems of this ”essentially diagonally dominant” form, reliable difference approximations can be generated and an error analysis carried out. The paper not only considers how to perform these tasks but also shows in detail how more general systems can be transformed into this form. A number of both linear and non-linear numerical examples are presented.