Python tools for the investigation of optimal explicit runge-kutta methods

Abstract

The investigation of many real-world applications involves mathematical models that consist of systems of ordinary differential equations (ODEs). This thesis mainly focuses on the ODEs with the initial values known as initial value ordinary differential equations. These equations are typically solved using numerical methods to obtain approximate solutions. A popular class of numerical methods to solve an initial value ODE are the Explicit Runge-Kutta (ERK) methods. This thesis considers Python software for the investigation of ERK methods. ERK methods can be used to obtain approximate solutions at a discrete set of points across the domain of interest, with ℎ being the distance between the points. An ERK method is said to be of order 𝑝 if the error of the numerical solution is proportional to ℎ𝑝 . In this thesis, we consider Python software for the determination of optimal ERK methods of orders 1 to 4. The Python software also has the capability to solve a set of test problems using various ERK methods in order to allow for a comparison of the accuracy of the numerical solutions obtained from the ERK methods. The Python software can also be used to extend the discrete approximate solutions from the ERK methods to obtain continuous approximate solutions over the entire domain using Hermite interpolation. To assess the accuracy of the continuous solution approximation, the software can also be used to compute the defect of the continuous approximate solution, where defect is the amount by which the continuous approximate solution fails to satisfy the ODE.