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.
