Runge-Kutta methods do not enjoy a very good reputation compared with multistep methods for computing ordinary differential equations, and so it was decided to attempt to improve their reliability and efficiency. To that end, in this thesis, five classes of generalized Runge-Kutta have been defined and designated as R(Jn), R(J̄n) R(Jn-k), P(Jn) and P(Jn-k) methods. The R(Jn), R(J̄n) and R(Jn-k) methods are (linearly) implicit and suitable for computing stiff systems. The P(Jn) and P(Jn-k) methods are explicit and suitable for computing non-stiff systems. The R(Jn) methods have received some attention in the literature, the R(J̄n) and P(Jn) methods have been only briefly considered and the R(Jn-k) and P(Jn-k) methods do not appear to have been considered previously. We have improved on the results available for processes of the type R(Jn), R(J̄n) and P(Jn), whilst the results for R(Jn-k) and P(Jn-k) methods are totally new.
The primary consideration in designing the processes has been reliability (that is, able to successfully compute a wide range of problems), rather than efficiency (that is, minimal computational work), and so the R(Jn), R(J̄n) and R(Jn-k) methods satisfy the demanding criterion of Internal S-Stability. A useful recursive formula for obtaining the stability functions has been obtained. The stability of the P(Jn) and P(Jn-k) methods is comparable to explicit Runge-Kuttas of the same order. Consistency conditions for processes of order less than or equal to four have been obtained. In order to obtain efficient processes (without sacrificing reliability), some results concerning the minimum number of function evaluations per step required for a process of particular order (for orders less than or equal to four) have been obtained.
In addition, an error estimator similar to that of Fehlberg or England has been developed. Although this error estimator is similar in spirit to the imbedded type formulas where a pth order process has a (p-1)th order process imbedded within it, such that subtraction of the two solutions gives the error estimate, it is shown that for these generalized Runge-Kuttas, the (p-1)th order process is not strictly "imbedded" within the pth order formula, but can be obtained from the information available from the computation of the pth order solution, with very little extra work. Thus it is a simple matter to obtain a cheap error estimator from any generalized Runge-Kutta process without having to simultaneously solve the nonlinear algebraic equations of consistency for orders p and (p-1). Several example processes of all five classes have been obtained and tested numerically.