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

翻译：本文提出了一种新的、更高效的单自由度问题数值方法。该方法基于论文第一部分所得的运动方程弱形式，采用分段多项式函数作为插值函数进行构造。其近似阶数可任意提高，与插值函数的次数成正比，仅受数值不稳定性制约。此外，系统的机械能保持守恒。因此，现有算法的所有显著缺陷——如Dahlqvist Barrier定理所施加的限制以及引入数值阻尼的必要性——均已得到解决。