New methods for oscillatory systems based on ARKN methods

DOI10.1016/j.apnum.2005.09.005zbMath1096.65068OpenAlexW2017098992MaRDI QIDQ2499747

J. M. Franco

Publication date: 14 August 2006

Published in: Applied Numerical Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.apnum.2005.09.005

zbMATH Keywords

stabilitynumerical examplesoscillatory systemsadapted Runge-Kutta-Nyström (ARKN) methodexplicit fourth-order methods

Mathematics Subject Classification ID

Nonlinear ordinary differential equations and systems (34A34) Stability and convergence of numerical methods for ordinary differential equations (65L20) Numerical methods for initial value problems involving ordinary differential equations (65L05) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06)

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Cites Work

• Unnamed Item
• Unnamed Item
• Runge-Kutta algorithms for oscillatory problems
• A 5(3) pair of explicit ARKN methods for the numerical integration of perturbed oscillators.
• On the numerical integration of orbital problems with high-order Runge-Kutta-Nyström methods
• New methods for oscillatory problems based on classical codes
• A Gautschi-type method for oscillatory second-order differential equations
• A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators
• Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators
• Families of Runge-Kutta-Nystrom Formulae
• Solving Ordinary Differential Equations I
• Diagonally Implicit Runge–Kutta–Nyström Methods for Oscillatory Problems
• Long-Time-Step Methods for Oscillatory Differential Equations
• On Magnus Integrators for Time-Dependent Schrödinger Equations
• Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations