Casimir preserving stochastic Lie–Poisson integrators

• Published in: Advances in continuous and discrete models Vol. 2024; no. 1; p. 1
• Main Authors: Luesink, Erwin, Ephrati, Sagy, Cifani, Paolo, Geurts, Bernard
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 02.01.2024; Springer Nature B.V
• Subjects: Algebra; Analysis; Difference and Functional Equations; Differential equations; Euler-Lagrange equation; Functional Analysis; Integrators; Lie groups; Mathematical analysis; Mathematics; Mathematics and Statistics; Mechanics; Methods; Numerical analysis; Numerical methods; Ordinary Differential Equations; Partial Differential Equations; Poisson equation; Runge-Kutta method; Time integration; Stochastic differential equations; 60H10; 65P10; Structure preservation; Geometric integration; 70G65; 70H99; 65C30; Hamiltonian mechanics; Lie group; Coadjoint orbits; Lie algebra; Stochastic Lie–Poisson integration
• ISSN: 2731-4235; 1687-1839; 2731-4235; 1687-1847
• DOI: 10.1186/s13662-023-03796-y

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.