Discrete Dynamics in Nature and Society 
Volume 2008 (2008), Article ID 790619, 38 pages
doi:10.1155/2008/790619
Research Article

Accelerated Runge-Kutta Methods

Firdaus  E. Udwadia1 and Artin Farahani2

 1Departments of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations Management, 430K Olin Hall, University of Southern California, Los Angeles, CA 90089-1453, USA
2Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, USA
Received 31 December 2007; Revised 8 March 2008; Accepted 27 April 2008

Recommended by Leonid Berezansky

Abstract

Standard Runge-Kutta methods are explicit, one-step, and generally constant step-size numerical integrators for the solution of initial value problems. Such integration schemes of orders 3, 4, and 5 require 3, 4, and 6 function evaluations per time step of integration, respectively. In this paper, we propose a set of simple, explicit, and constant step-size Accerelated-Runge-Kutta methods that are two-step in nature. For orders 3, 4, and 5, they require only 2, 3, and 5 function evaluations per time step, respectively. Therefore, they are more computationally efficient at achieving the same order of local accuracy. We present here the derivation and optimization of these accelerated integration methods. We include the proof of convergence and stability under certain conditions as well as stability regions for finite step sizes. Several numerical examples are provided to illustrate the accuracy, stability, and efficiency of the proposed methods in comparison with standard Runge-Kutta methods.