Mixed collocation methods for y” = f(x , y)

Abstract

The second-order initial value problem y" = f(x,y), y(x(_0)) = y(_0), y'(x(_0)) = z(_0) which does not contain the first derivative explicitly and where the solution is oscillatory has been of great interest for many years. Our aim is to construct numerical methods which are tuned to act efficiently on strongly oscillating functions. The frequencies involved determine the oscillatory character of the function and as the frequencies approach zero, the classical methods are obtained. The exponential- fitting tool has become increasingly popular as it is specially tailored for oscillating functions. Many classes of methods have been used with exponential-fitting and this will be discussed in more detail in the thesis. Collocation methods are considered for which the basis functions are combinations of polynomial and trigonometric terms. The resulting methods can be regarded as Runge-Kutta-Nyström methods with steplength dependent coefficients. We show how order conditions may be obtained, investigate the stability and other properties of particular methods and present some numerical results.