An Essential Extension of the Finite-Energy Condition for Extended Runge-Kutta-Nyström Integrators when Applied to Nonlinear Wave Equations (Communications in Computational Physics, 2017)
This paper is devoted to an extension of the finite-energy condition for extendedRunge-Kutta-Nyström (ERKN) integrators and applications to nonlinear waveequations. We begin with an error analysis for the integrators for multi-frequencyhighly oscillatory systems $y′′+My = f(y)$, where $M$ is positive semi-definite, ║$M$║≫║$\frac{∂f}{∂y}$║, and ║$M$║≫1. The highly oscillatory system is due to the semi-discretisation ofconservative, or dissipative, nonlinear wave equations. The structure of such a matrix $M$ and initial conditions are based on particular spatial discretisations. Similarly tothe error analysis for Gaustchi-type methods of order two, where a finite-energy conditionbounding amplitudes of high oscillations is satisfied by the solution, a finite-energycondition for the semi-discretisation of nonlinear wave equations is introducedand analysed. These ensure that the error bound of ERKN methods is independentof ║$M$║. Since stepsizes are not restricted by frequencies of $M$, large stepsizes can beemployed by our ERKN integrators of arbitrary high order. Numerical experimentsprovided in this paper have demonstrated that our results are truly promising, andconsistent with our analysis and prediction.