Efficient iterations for Gauss methods on second-order problems

We consider some important aspects about the implementation of high order implicit formulas (specially the Gauss methods) for solving second-order differential systems having high frequencies and small amplitudes superimposed. The choice of an appropriate iterative scheme is discussed in detail. Important topics about the predictors (initial guesses) are analyzed and a variable order strategy to select the best predictor at each integration step is supplied. A few numerical experiments on some standard test problems confirm the theory presented.