@Article{IJNAM-4-2,
author = {},
title = {Variable Step-Size Selection Methods for Implicit Integration Schemes for ODEs},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2007},
volume = {4},
number = {2},
pages = {210--240},
abstract = {<p style="text-align: justify;">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&#39;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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2007-IJNAM-860},
url = {https://global-sci.com/article/83767/variable-step-size-selection-methods-for-implicit-integration-schemes-for-odes}
}