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Numerical integrations of stochastic contact Hamiltonian systems via stochastic contact Hamilton-Jacobi equation

Qingyi Zhan, Jinqiao Duan, Xiaofan Li, Lijin Wang

Abstract

Stochastic contact Hamiltonian systems are a class of important mathematical models, which can describe the dissipative properties with odd dimensions in the stochastic environment. In this article, we investigate the numerical dynamics of the stochastic contact Hamiltonian systems via structure-preserving methods. The contact structure-preserving schemes are constructed by the stochastic contact Hamilton-Jacobi equation. A general numerical approximation method of the stochastic contact Hamilton-Jacobi equation is devised, and the convergent order theorem is provided, too. Numerical tests are shown to confirm the theoretical results and the usability of proposed approach.

Numerical integrations of stochastic contact Hamiltonian systems via stochastic contact Hamilton-Jacobi equation

Abstract

Stochastic contact Hamiltonian systems are a class of important mathematical models, which can describe the dissipative properties with odd dimensions in the stochastic environment. In this article, we investigate the numerical dynamics of the stochastic contact Hamiltonian systems via structure-preserving methods. The contact structure-preserving schemes are constructed by the stochastic contact Hamilton-Jacobi equation. A general numerical approximation method of the stochastic contact Hamilton-Jacobi equation is devised, and the convergent order theorem is provided, too. Numerical tests are shown to confirm the theoretical results and the usability of proposed approach.

Paper Structure

This paper contains 12 sections, 6 theorems, 93 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic contact Hamilton-Jacobi Theory
  4. Numerical scheme via the stochastic contact Hamilton-Jacobi equation
  5. The construction of the contact scheme
  6. The construction of the contact Euler scheme of order 1.0
  7. Convergence Analysis
  8. Numerical Experiments
  9. Construction of contact structure-preserving schemes
  10. Numerical implementations of contact structure-preserving schemes
  11. Numerical results
  12. Conclusion

Key Result

Theorem 3.1

Let $S(Q,C,t)$ satisfy the stochastic contact Hamilton-Jacobi equation in the form of where $C=(C_1,C_2,...,C_n)$ is constant, and suppose If there exists a stopping time $\tau>t_0$ a.s., then for almost every $\omega\in \Omega$, for each $t\in[t_0,\tau)$, the mapping $(q,p,s)\rightarrow (Q(t),P(t),S(Q,C,t))$ given by $(3.3)$ is the phase flow of the stochastic contact Hamiltonian systems $(2.1)

Figures (5)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • Theorem 5.4
  • ...and 1 more