[Paper Review] Long-time numerical energy conservation of extended RKN integrators for highly oscillatory Hamiltonian systems

Bin Wang, Xinyuan Wu|arXiv (Cornell University)|Dec 13, 2017

Numerical methods for differential equations24 references3 citations

TL;DR

This paper analyzes long-time energy conservation in extended RKN (ERKN) integrators for highly oscillatory Hamiltonian systems using modulated Fourier expansion. It demonstrates that symmetric ERKN methods preserve both total and oscillatory energy nearly exactly over long times, establishing two almost-invariants crucial for numerical stability in highly oscillatory dynamics.

ABSTRACT

The aim of this paper is to analyze the long-time behaviour of extended Runge--Kutta--Nystr\{o}m (ERKN) integrators for highly oscillatory Hamiltonian systems. It is important to note that the well-known Gautschi-type methods of order two yield examples of ERKN integrators. To this end, we use the modulated Fourier expansion to analyze the long-time numerical energy conservation. We will show that the ERKN integrators with symmetry conditions have two almost-invariants and have a near conservation of the total and oscillatory energy over a long term.

Motivation & Objective

• To investigate the long-time numerical energy behavior of extended RKN (ERKN) integrators in highly oscillatory Hamiltonian systems.
• To establish the existence of almost-invariants in symmetric ERKN methods.
• To analyze the near-conservation of total and oscillatory energy over long integration intervals.
• To validate that Gautschi-type methods of order two are special cases of ERKN integrators.
• To provide a theoretical foundation for energy preservation using modulated Fourier expansion.

Proposed method

• Utilizes modulated Fourier expansion to analyze the long-time behavior of ERKN integrators.
• Applies symmetry conditions to ERKN methods to derive structural properties.
• Derives two almost-invariants from the modulated Fourier expansion for symmetric ERKN schemes.
• Analyzes the near-conservation of total and oscillatory energy components over long-term integration.
• Compares ERKN methods with known Gautschi-type methods to confirm their inclusion as special cases.
• Establishes theoretical bounds on energy deviation using asymptotic expansion techniques.

Experimental results

Research questions

• RQ1 Do symmetric ERKN integrators exhibit long-term near-conservation of total and oscillatory energy in highly oscillatory Hamiltonian systems?
• RQ2 What almost-invariants emerge in symmetric ERKN methods under modulated Fourier analysis?
• RQ3 How do Gautschi-type methods of order two relate to the broader class of ERKN integrators?
• RQ4 To what extent do ERKN integrators preserve energy over long integration intervals?
• RQ5 What is the role of symmetry in ensuring long-term energy conservation in ERKN schemes?

Key findings

• Symmetric ERKN integrators possess two almost-invariants that govern long-term energy behavior.
• The total energy and oscillatory energy components are nearly conserved over long integration times.
• The modulated Fourier expansion reveals that energy deviations grow slowly, indicating near-conservation.
• Gautschi-type methods of order two are confirmed as specific instances of ERKN integrators.
• The theoretical framework establishes that symmetric ERKN methods maintain energy structure over extended periods.
• The analysis confirms that long-time energy conservation is a direct consequence of symmetry and the modulated Fourier approach.