Second derivative methods applied to implicit first- and second-order systems

A wide variety of methods have been proposed for solving implicit second-order systems of differential equations such as arise in structural and mechanical vibration analyses. Here we discuss a class of methods, based on second derivative formulae, some of which have previously been used by Enright.¹ We show them to have desirable analytic and implementation properties. We relate these methods to others recently proposed by Brusa and Nigro and by Serbin, showing that we obtain algorithms which have all the desirable properties of their methods and which can also be extended to classes of practical problems which were not considered by these other authors. Since implicit first-order systems are also important in practice, for example in finite element analyses of heat conduction problems, we demonstrate the efficient implementation of the second derivative formulae in this case too.