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Open AccessFeature PaperArticle

Locally Exact Integrators for the Duffing Equation

Wydział Fizyki, Uniwersytet w Białymstoku, ul. Ciołkowskiego 1L, 15-245 Białystok, Poland
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Mathematics 2020, 8(2), 231; https://doi.org/10.3390/math8020231
Received: 17 January 2020 / Revised: 1 February 2020 / Accepted: 4 February 2020 / Published: 10 February 2020
(This article belongs to the Special Issue Geometric Numerical Integration)
A numerical scheme is said to be locally exact if after linearization (around any point) it becomes exact. In this paper, we begin with a short review on exact and locally exact integrators for ordinary differential equations. Then, we extend our approach on equations represented in the so called linear gradient form, including dissipative systems. Finally, we apply this approach to the Duffing equation with a linear damping and without external forcing. The locally exact modification of the discrete gradient scheme preserves the monotonicity of the Lyapunov function of the discretized equation and is shown to be very accurate.
Keywords: geometric numerical integration; exact discretization; locally exact methods; discrete gradient method; linear gradient form of ODEs; dissipative systems geometric numerical integration; exact discretization; locally exact methods; discrete gradient method; linear gradient form of ODEs; dissipative systems
MDPI and ACS Style

Cieśliński, J.L.; Kobus, A. Locally Exact Integrators for the Duffing Equation. Mathematics 2020, 8, 231.

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