Analysis of Runge-Kutta methods using Butcher tableaus

Abstract

This Bachelor thesis provides an analysis of Runge-Kutta methods using Butcher tableaus. Runge-Kutta method are numerical methods used for approximating initial value problems. A Runge-Kutta method can be classified as either an explicit or an implicit method. A special kind of implicit methods are diagonally implicit methods. The type of method can be recognised by the Butcher tableau.

Using the entries of the Butcher tableau, one can compute the amplification factor of a Runge-Kutta method. The amplification factor can then be used to compute the order of the local truncation error and the stability region. Examples of these computations are given for seven methods. Furthermore, this thesis provides an algorithm to perform time steps for each of the three types of Runge-Kutta methods. Finally, in order to analyse the global truncation error of the seven methods, the algorithm to perform time steps is used with different step sizes.