Decoupling and stability of algorithms for boundary value problems

The ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes. Therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals. It is shown that most methods (like multiple shooting, collocation, invariant imbedding and difference methods) derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later (for instance when solving a linear system). This decoupling is related to the dichotomy of the ordinary differential equations. In fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed. How this stability is related to the use of the boundary conditions is also investigated.