The Numerical Solution of Differential Equations by Third and Fourth Order Runge-Kutta Methods

Abstract

C. Runge originally suggested the numerical methods of solving differential equations which will be examined, and were subsequently improved on by, to mention a few, K.Heun, and W. Kutta. The entirety of these methods have, as a result, been referred to as the Runge-Kutta methods for the numerical solution of differential equations. The first section of the thesis consists of the derivation of third and fourth order Runge-Kutta methods and their respective truncation errors. Notation, definitions, and various concepts are introduced as needed in the various sections. The numerical solutions of differential equations using third order Runge-Kutta methods are then discussed in the second section. Various formulae and relationships are derived here for third order methods. In all numerical tables that follow, the results were obtained using a Bendix Model G-15 Digital Computer. In the third section, one considers fourth order Runge-Kutta methods for the numerical solution of ordinary differential equations. However, in addition to considerations of symmetry, reduction of operations and storage requirements, as examined in section two, one examines a Runge-Kutta method due to Blum which basically modifies a programming procedure. Finally in the last section, one investigates methods due to A. Ralston which minimize a bound on the truncation error derived in the first section. An appendix is also included containing various programs for the Bendix G-15D that have been needed throughout the sections.