Casimir preserving stochastic Lie–Poisson integrators

Erwin Luesink; Sagy Ephrati; Paolo Cifani; Bernard Geurts

doi:10.1186/s13662-023-03796-y

Casimir preserving stochastic Lie–Poisson integrators

Erwin Luesink^*, Sagy Ephrati, Paolo Cifani, Bernard Geurts

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

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Abstract

Casimir preserving integrators for stochastic Lie–Poisson equations with Stratonovich noise are developed, extending Runge–Kutta Munthe-Kaas methods. The underlying Lie–Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie–Poisson dynamics using the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations.

Original language	English
Article number	1
Journal	Advances in Continuous and Discrete Models
Volume	2024
Issue number	1
Early online date	2 Jan 2024
DOIs	https://doi.org/10.1186/s13662-023-03796-y
Publication status	Published - Dec 2024

Keywords

Coadjoint orbits
Geometric integration
Hamiltonian mechanics
Lie algebra
Lie group
Stochastic differential equations
Stochastic Lie–Poisson integration
Structure preservation

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title = "Casimir preserving stochastic Lie–Poisson integrators",

abstract = "Casimir preserving integrators for stochastic Lie–Poisson equations with Stratonovich noise are developed, extending Runge–Kutta Munthe-Kaas methods. The underlying Lie–Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie–Poisson dynamics using the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations.",

keywords = "Coadjoint orbits, Geometric integration, Hamiltonian mechanics, Lie algebra, Lie group, Stochastic differential equations, Stochastic Lie–Poisson integration, Structure preservation",

author = "Erwin Luesink and Sagy Ephrati and Paolo Cifani and Bernard Geurts",

note = "Publisher Copyright: {\textcopyright} 2023, The Author(s). Financial transaction number: 2500115063",

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T1 - Casimir preserving stochastic Lie–Poisson integrators

AU - Luesink, Erwin

AU - Ephrati, Sagy

AU - Cifani, Paolo

AU - Geurts, Bernard

PY - 2024/12

Y1 - 2024/12

N2 - Casimir preserving integrators for stochastic Lie–Poisson equations with Stratonovich noise are developed, extending Runge–Kutta Munthe-Kaas methods. The underlying Lie–Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie–Poisson dynamics using the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations.

AB - Casimir preserving integrators for stochastic Lie–Poisson equations with Stratonovich noise are developed, extending Runge–Kutta Munthe-Kaas methods. The underlying Lie–Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie–Poisson dynamics using the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations.

KW - Coadjoint orbits

KW - Geometric integration

KW - Hamiltonian mechanics

KW - Lie algebra

KW - Lie group

KW - Stochastic differential equations

KW - Stochastic Lie–Poisson integration

KW - Structure preservation

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DO - 10.1186/s13662-023-03796-y

M3 - Article

AN - SCOPUS:85181205144

SN - 2731-4235

VL - 2024

JO - Advances in Continuous and Discrete Models

JF - Advances in Continuous and Discrete Models

IS - 1

M1 - 1

ER -

Casimir preserving stochastic Lie–Poisson integrators

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