Stability concepts in the numerical solution of difference and differential equations

AbstractSeveral algorithms have been proposed for the stable numerical computation of non-dominant solutions of linear difference equations. The unifying approach of these algorithms is that they replace the task of solving the given initial value problem, by that of solving a boundary value problem, which is in some sense, equivalent. A similar task is involved in the numerical solution of initial value problems for ordinary differential equations using linear multistep methods. However, in this case, it is always assumed that the numerical solution will be found by forward recurrence, and this assumption imposes severe restrictions on the numerical methods that can be used. By adapting some of the ideas that have been proposed for linear recurrence relations, new algorithms are derived for the numerical solution of initial value problems, and it is shown that these new methods can be competitive with the best methods currently available