@Article{CiCP-22-3,
author = {},
title = {An Essential Extension of the Finite-Energy Condition for Extended Runge-Kutta-Nyström Integrators when Applied to Nonlinear Wave Equations},
journal = {Communications in Computational Physics},
year = {2017},
volume = {22},
number = {3},
pages = {742--764},
abstract = {<p style="text-align: justify;">This paper is devoted to an extension of the finite-energy condition for extended
Runge-Kutta-Nyström (ERKN) integrators and applications to nonlinear wave
equations. We begin with an error analysis for the integrators for multi-frequency
highly oscillatory systems $y′′+My = f(y)$, where $M$ is positive semi-definite, ║$M$║≫<span style="text-align: justify;">║</span>$\frac{∂f}{∂y}$<span style="text-align: justify;">║</span>, and ║$M$║≫1. The highly oscillatory system is due to the semi-discretisation of
conservative, or dissipative, nonlinear wave equations. The structure of such a matrix $M$ and initial conditions are based on particular spatial discretisations. Similarly to
the error analysis for Gaustchi-type methods of order two, where a finite-energy condition
bounding amplitudes of high oscillations is satisfied by the solution, a finite-energy
condition for the semi-discretisation of nonlinear wave equations is introduced
and analysed. These ensure that the error bound of ERKN methods is independent
of ║$M$║. Since stepsizes are not restricted by frequencies of $M$, large stepsizes can be
employed by our ERKN integrators of arbitrary high order. Numerical experiments
provided in this paper have demonstrated that our results are truly promising, and
consistent with our analysis and prediction.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2016-0141},
url = {https://global-sci.com/article/80158/an-essential-extension-of-the-finite-energy-condition-for-extended-runge-kutta-nystrom-integrators-when-applied-to-nonlinear-wave-equations}
}