Harmonic-Enriched Reproducing Kernel Approximation for Highly Oscillatory Differential Equations

2020 ◽  
Vol 146 (4) ◽  
pp. 04020014 ◽  
Author(s):  
Ashkan Mahdavi ◽  
Sheng-Wei Chi ◽  
Negar Kamali
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Kernel Approximation ◽  
Highly Oscillatory ◽  
Highly Oscillatory Differential Equations

scholarly journals RECONSIDERING TRIGONOMETRIC INTEGRATORS

2009 ◽  
Vol 50 (3) ◽  
pp. 320-332 ◽  
Author(s):  
DION R. J. O’NEALE ◽  
ROBERT I. MCLACHLAN
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Nonlinear Stability ◽  
Higher Order ◽  
Statistical Properties ◽  
Highly Oscillatory ◽  
Time Averages ◽  
Highly Oscillatory Differential Equations ◽  
Low Order

AbstractIn this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi–Pasta–Ulam problem, a highly oscillatory Hamiltonian system. We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but at the wrong values. We use statistical properties such as time averages to further evaluate the performance of the trigonometric methods and compare the performance with that of the mid-point rule.

scholarly journals Multi-revolution composition methods for highly oscillatory differential equations

2014 ◽  
Vol 128 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Philippe Chartier ◽  
Joseba Makazaga ◽  
Ander Murua ◽  
Gilles Vilmart
Keyword(s):  
Differential Equations ◽  
Composition Methods ◽  
Highly Oscillatory ◽  
Highly Oscillatory Differential Equations

scholarly journals Word series high-order averaging of highly oscillatory differential equations with delay

2019 ◽  
Vol 4 (2) ◽  
pp. 445-454 ◽  
Author(s):  
J. M. Sanz-Serna ◽  
Beibei Zhu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
High Order ◽  
Constant Delay ◽  
Integer Multiple ◽  
Averaged Equations ◽  
Highly Oscillatory ◽  
Highly Oscillatory Differential Equations ◽  
Delay Differential ◽  
Equations With Delay

AbstractWe show that, when the delay is an integer multiple of the forcing period, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay. Our approach is based on the use of word series techniques to obtain high-order averaged equations for differential equations without delay.

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