Some investigations into the numerical solution of initial value problems in ordinary differential equations

Abstract

In this thesis several topics in the numerical solution of the initial value problem in first-order ordinary diff'erentlal equations are investigated, Consideration is given initially to stiff differential equations and their solution by stiffly-stable linear multistep methods which incorporate second derivative terms. Attempts are made to increase the size of the stability regions for these methods both by particular choices for the third characteristic polynomial and by the use of optimization techniques while investigations are carried out regarding the capabilities of a high order method. Subsequent work is concerned with the development of Runge-Kutta methods which include second-derivative terms and are implicit with respect to y rather than k. Methods of order three and four are proposed which are L-stable. The major part of the thesis is devoted to the establishment of recurrence relations for operators associated with linear multistep methods which are based on a non-polynomial representation of the theoretical solution. A complete set of recurrence relations is developed for both implicit and explicit multistep methods which are based on a representation involving a polynomial part and any number of arbitrary functions. The amount of work involved in obtaining mulc iste, :ne::l'lJds by this technique is considered and criteria are proposed to Jecide when this particular method of derivation should be em~loyed. The thesis is conclud~d by using Prony's method to develop one-step methods and multistep methods which are exponentially adaptive and as such can be useful in obtaining solutions to problems which are exponential in nature.