A study of continuous explicit and diagonally implicit Runge-Kutta methods

Runge-Kutta methods provide a popular way to solve ordinary differential equations. However, traditional schemes only provide solutions at a discrete set of points. In this thesis, continuous Runge-Kutta methods for first order ordinary differential equations are considered. The developments of some continuous explicit Runge-Kutta formulas by various researchers are presented in the first chapter. They impose different conditions on their schemes, but all make use of what are called order conditions. In the second chapter, two-stage and three-stage continuous diagonally implicit methods are developed. With the two-stage continuous method, second order may be attained with third order at the mesh points. For the particular third-order continuous method presented, third order may be attained with fourth order at the mesh points. As revealed in the conclusion of the thesis, a three-stage continuous diagonally implicit Runge-Kutta method that is globally third order and fourth order at the mesh points will not satisfy certain stability requirements.