On the numerical solutions of differential equations with time-stepping methods

In this thesis, algorithms based on one- and two-step methods have been given for the numerical integration of ordinary differential equations in the form of $y'=f(y)$. Classical Runge-Kutta methods are one-step methods used commonly for the numerical solution of problems in this form. Recently, studies have been done to improve the accuracy and order of these methods. A part of these studies consists of the approaches including higher order derivatives of the solution. In line with this purpose, three-derivative Runge-Kutta methods including the second and third derivatives of the solution have been given for algorithms based on one-step methods. In case of algorithms based on two-step methods, two-step Runge-Kutta methods have been extended to two-derivative two-step Runge-Kutta methods including the second derivative of the solution. In this study, Chan and Chan's approach has been followed which based on the algebraic theory of Butcher and the $\mathcal{B}$-series theory of Hairer and Wanner for order conditions of both methods. Later, the stability analysis of the methods has been done. Some standard test problems have been tested for the performance of these methods and made comparisons with the methods in the literature.