Runge-Kutta methods : analysis and implementation /

Abstract

In this thesis, we have developed new numerical methods in Runge-Kutta family for numerical solution of ordinary di erential equations. We have extended the idea of e ective order to Runge-Kutta Nystr om methods for numerical approximation of second order ordinary di erential equations. The composition of Runge-Kutta Nystr om methods, the pruning of associated Nystr om trees, and conditions for e ective order Runge-Kutta Nystr om methods up to order ve are presented. Also, partitioned Runge-Kutta methods of e ective order 4 with 3 stages are constructed. The most obvious feature of these methods is e ciency in terms of implementation cost. The numerical results verify that the asymptotic error behavior of the e ective order 4 partitioned Runge- Kutta methods with 3 stages is similar to that of classical order 4 method which necessarily require 4 stages. Moreover, it is evident from the numerical results that e ective order methods are more e cient than their classical order counterpart. Lastly, a family of explicit symplectic partitioned Runge-Kutta methods are derived with e ective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more e cient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian system con rming the e ciency of the approach is also provided with good energy conservation.