  A new, more efficient, numerical method for the SDOF problem is presented.
Its construction is based on the weak form of the equation of motion, as
obtained in part I of the paper, using piece-wise polynomial functions as
interpolation functions. The approximation rate can be arbitrarily high,
proportional to the degree of the interpolation functions, tempered only by
numerical instability. Moreover, the mechanical energy of the system is
conserved. Consequently, all significant drawbacks of existing algorithms, such
as the limitations imposed by the Dahlqvist Barrier theorem and the need for
introduction of numerical damping, have been overcome.
