Abstract:
The methods studied in this thesis for the numerical solution of ordinary differential equations are a special case of multi-derivative Runge-Kutta methods first investigated by Kastlunger and Wanner. We have restricted our investigation to only those methods that involve the first and second derivatives called two-derivative Runge-Kutta methods or TDRK for short. The study is motivated by the observation that TDRK methods of a given order can be constructed with fewer function evaluations compared to Runge-Kutta methods. This raises the possibility of cheaper implementation costs. An additional motivation is that the TDRK methods can be constructed with high stage order. This important property makes these methods potentially useful for the solution of stiff problems. We derived order conditions and constructed a variety of TDRK methods having various orders. We made experimental comparisons of these methods among themselves and with other Runge-Kutta methods on both linear and nonlinear problems. The explicit methods also include embedded TDRK methods for experiments with variable stepsize. A feature of the explicit methods is that, unlike Runge-Kutta methods, the TDRK methods can attain stage order 2 and are shown to have a marginal advantage in solving mildly stiff problems. For stiff problems we constructed a variety of A-stable and L-stable methods and compared these TDRK methods with some popular implicit Runge-Kutta methods. We find that some TDRK methods are significantly more efficient than the Runge-Kutta methods of the same order. The results indicate particularly good performance for two of the order-4, semi-implicit, A-stable TDRK methods. Both these methods are often more efficient than some other higher order methods such as the 3-stage Radau IIA. For stiff problems with perturbed initial values we confirm that L-stable methods are superior to A-stable methods. For such cases the order-5 L-stable TDRK method was more efficient than the other methods tested. We have also extended the application of TDRK methods to partial differential equations. The introduction of the second derivative leads to a new approach to semi-discretize these equations. In particular, we showed that an order-4 implicit TDRK method is significantly more efficient than the popular Crank-Nicolson method for diffusion equations. Finally this thesis has provided good evidence to suggest that these TDRK methods have potential as viable methods for the numerical solution of certain ordinary and partial differential equations.