Methods without saturation for boundary value problems

We consider a class of numerical methods without saturation for boundary value problems which is oriented for computations with an arbitrary controlled precision. These methods differ from the traditional ones in the feedback which they have with the problem, when the results of computations allow to draw conclusions on analytical properties of solutions to the problem. These methods are little-known, and they are not implemented in existing computer algebra systems, although these systems made their practical realization possible.