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Structure-Preserving Implicit Runge-Kutta Methods for Stochastic Poisson Systems with Multiple Noises

Liying Zhang, Fenglin Xue, Lijin Wang

TL;DR

The paper tackles structure preservation for stochastic Poisson systems driven by multiple noises and introduces two families of implicit Runge-Kutta methods. It develops transformed Runge-Kutta methods via a Darboux–Lie coordinate change to handle non-constant structure matrices and diagonal implicit Runge-Kutta methods for constant structure matrices, proving preservation of the Poisson structure, Casimir invariants, and quadratic Hamiltonians where appropriate. Numerical experiments on stochastic rigid body dynamics and linear stochastic Poisson systems confirm invariant preservation and mean-square order 1, demonstrating long-term stability and accuracy. These methods extend the toolkit for structure-preserving integration in stochastic Poisson dynamics and suggest avenues for optimization and SPDE extensions.

Abstract

In this paper, we propose the diagonal implicit Runge-Kutta methods and transformed Runge-Kutta methods for stochastic Poisson systems with multiple noises. We prove that the first methods can preserve the Poisson structure, Casimir functions, and quadratic Hamiltonian functions in the case of constant structure matrix. Darboux-Lie theorem combined with coordinate transformation is used to construct the transformed Runge-Kutta methods for the case of non-constant structure matrix that preserve both the Poisson structure and the Casimir functions. Finally, through numerical experiments on stochastic rigid body systems and linear stochastic Poisson systems, the structure-preserving properties of the proposed two kinds of numerical methods are effectively verified.

Structure-Preserving Implicit Runge-Kutta Methods for Stochastic Poisson Systems with Multiple Noises

TL;DR

The paper tackles structure preservation for stochastic Poisson systems driven by multiple noises and introduces two families of implicit Runge-Kutta methods. It develops transformed Runge-Kutta methods via a Darboux–Lie coordinate change to handle non-constant structure matrices and diagonal implicit Runge-Kutta methods for constant structure matrices, proving preservation of the Poisson structure, Casimir invariants, and quadratic Hamiltonians where appropriate. Numerical experiments on stochastic rigid body dynamics and linear stochastic Poisson systems confirm invariant preservation and mean-square order 1, demonstrating long-term stability and accuracy. These methods extend the toolkit for structure-preserving integration in stochastic Poisson dynamics and suggest avenues for optimization and SPDE extensions.

Abstract

In this paper, we propose the diagonal implicit Runge-Kutta methods and transformed Runge-Kutta methods for stochastic Poisson systems with multiple noises. We prove that the first methods can preserve the Poisson structure, Casimir functions, and quadratic Hamiltonian functions in the case of constant structure matrix. Darboux-Lie theorem combined with coordinate transformation is used to construct the transformed Runge-Kutta methods for the case of non-constant structure matrix that preserve both the Poisson structure and the Casimir functions. Finally, through numerical experiments on stochastic rigid body systems and linear stochastic Poisson systems, the structure-preserving properties of the proposed two kinds of numerical methods are effectively verified.
Paper Structure (10 sections, 8 theorems, 53 equations, 9 figures)

This paper contains 10 sections, 8 theorems, 53 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Stochastic Poisson systems
  3. Runge-Kutta methods for stochastic Poisson systems
  4. Transformed Runge-Kutta methods
  5. Diagonal implicit Runge-Kutta methods
  6. Main results
  7. Numerical experiments
  8. Stochastic rigid body systems
  9. Linear stochastic Poisson systems
  10. Conclusion

Key Result

Theorem 2.1

(Fengetal2010) For a Poisson manifold with structure matrix $B(y)$, 2.5 is equivalent to

Figures (9)

  • Figure 1: Sample paths produced by \ref{['5.5']} and \ref{['5.6']}.
  • Figure 2: Phase orbits produced by \ref{['5.5']} and \ref{['5.6']}.
  • Figure 3: Casimir functions produced by \ref{['5.5']} and \ref{['5.6']}.
  • Figure 4: Mean-square order of the transformed Runge-Kutta methods.
  • Figure 5: Sample paths produced by \ref{['5.9']} and \ref{['5.10']}.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Definition 1
  • Theorem 2.1
  • Definition 2
  • Theorem 3.1
  • Theorem 3.2: Darboux-Lie theorem
  • Remark 1
  • Theorem 3.3
  • proof
  • Remark 2
  • Theorem 4.1
  • ...and 7 more