Conservative spline methods for second-order IVPs of ODEs

One of the most fundamental characteristics in physics are the conservation laws [1]. Therefore, the numerical methods in computational physics should be conservative. Unfortunately, the most popular numerical methods, like Runge-Kutta, multistep, and Taylor expansion methods, when they are applied to the second-order, nonlinear, Newtonian initial value problems, do not preserve fundamental physical invariants, like energy and momentum, when these are present. In [2–4], a family of spline methods proposed for the initial value problems of order n. A variant of these methods are energy-conserving approximations of order three, four, and five for the second-order Newtonian initial value problems. Furthermore, it seems to be easy to extend for equations containing the first derivative of the unknown function (even for coupled equations). Some applications and numerical results are presented. © 1999 Elsevier Science Ltd. All rights reserved.