- Journal Home
- Volume 22 - 2025
- Volume 21 - 2024
- Volume 20 - 2023
- Volume 19 - 2022
- Volume 18 - 2021
- Volume 17 - 2020
- Volume 16 - 2019
- Volume 15 - 2018
- Volume 14 - 2017
- Volume 13 - 2016
- Volume 12 - 2015
- Volume 11 - 2014
- Volume 10 - 2013
- Volume 9 - 2012
- Volume 8 - 2011
- Volume 7 - 2010
- Volume 6 - 2009
- Volume 5 - 2008
- Volume 4 - 2007
- Volume 3 - 2006
- Volume 2 - 2005
- Volume 1 - 2004
Cited by
- BibTex
- RIS
- TXT
Implicit integration schemes for ODEs, such as Runge-Kutta and Runge-Kutta-Nyström methods, are widely used in mathematics and engineering to numerically solve ordinary differential equations. Every integration method requires one to choose a step-size, $h$, for the integration. If $h$ is too large or too small the efficiency of an implicit scheme is relatively low. As every implicit integration scheme has a global error inherent to the scheme, we choose the total number of computations in order to achieve a prescribed global error as a measure of efficiency of the integration scheme. In this paper, we propose the idea of choosing $h$ by minimizing an efficiency function for general Runge-Kutta and Runge-Kutta-Nyström integration routines. This efficiency function is the critical component in making these methods variable step-size methods. We also investigate solving the intermediate stage values of these routines using both Newton's method and Picard iteration. We then show the efficacy of this approach on some standard problems found in the literature, including a well-known stiff system.
}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/860.html} }Implicit integration schemes for ODEs, such as Runge-Kutta and Runge-Kutta-Nyström methods, are widely used in mathematics and engineering to numerically solve ordinary differential equations. Every integration method requires one to choose a step-size, $h$, for the integration. If $h$ is too large or too small the efficiency of an implicit scheme is relatively low. As every implicit integration scheme has a global error inherent to the scheme, we choose the total number of computations in order to achieve a prescribed global error as a measure of efficiency of the integration scheme. In this paper, we propose the idea of choosing $h$ by minimizing an efficiency function for general Runge-Kutta and Runge-Kutta-Nyström integration routines. This efficiency function is the critical component in making these methods variable step-size methods. We also investigate solving the intermediate stage values of these routines using both Newton's method and Picard iteration. We then show the efficacy of this approach on some standard problems found in the literature, including a well-known stiff system.