ABSTRACT
In this chapter, the authors present various methods of approximation for obtaining numerical solutions of ordinary differential equations (ODEs) in time resulting from the space-time decoupling of initial value problems (IVPs). In order to obtain numerical solutions of the IVP, one now must employ methods of approximation to these ODEs in time. This process will yield numerical values of the nodal degrees of freedom for the entire time history. An approach to construct integral form from the ODEs in time is to use the residual functional resulting from the ODEs in time. Runge–Kutta methods of various orders are techniques of approximating ▵ φn over the interval [tn,tn+1] $ [t_{n} , t_{n + 1} ] $. The authors consider computation of evolution for some initial value model problems using central difference method, Houbolt method, Wilson's θ method, and Newmark's method for the ODEs in time resulting from decoupling of space and time using finite element approximation in spatial domain.
