Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems

Lijin Wang; Pengjun Wang; Yanzhao Cao

doi:10.3934/dcdss.2021095

Article Contents

2022, Volume 15, Issue 4: 819-836. Doi: 10.3934/dcdss.2021095

This issue Previous Article Bayesian topological signal processing Next Article Resampled ensemble Kalman inversion for Bayesian parameter estimation with sequential data

Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems

1.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
2.
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

^* Corresponding author: Yanzhao Cao
^* Corresponding author: Yanzhao Cao

Received: November 30, 2020

Revised: June 30, 2021

Early access: August 2021

Published: April 2022

The first and second authors are supported by NNSFC No. 11971458, No. 11471310

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Abstract

In this paper, we propose a class of numerical schemes for stochastic Poisson systems with multiple invariant Hamiltonians. The method is based on the average vector field discrete gradient and an orthogonal projection technique. The proposed schemes preserve all the invariant Hamiltonians of the stochastic Poisson systems simultaneously, with possibility of achieving high convergence orders in the meantime. We also prove that our numerical schemes preserve the Casimir functions of the systems under certain conditions. Numerical experiments verify the theoretical results and illustrate the effectiveness of our schemes.

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Mathematics Subject Classification: 60H35, 60H15, 65C30, 60H10, 65D30.

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Figure 1. Root mean-square convergence orders of the Milstein scheme, the Klöden scheme, the P-Milstein scheme, and the P-Klöden scheme

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Figure 2. Evolution of $ H^0(y),H^1(y) $ by the Milstein scheme and the P-Milstein scheme for system (32)

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Figure 3. Evolution of $ y^1 $ by the Milstein scheme and the P-Milstein scheme

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Figure 4. Evolution of the Casimir function by the Milstein scheme and the P-Milstein scheme

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Figure 5. Root mean-square convergence orders of the Euler scheme, the Milstein scheme, the P-Euler scheme, and the P-Milstein scheme

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Figure 6. Evolution of $ H^0(X),\,\,H^1(X) $ by the Milstein scheme and the P-Milstein scheme for the system (34)

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Figure 7. A sample path of $ y^1 $ produced by the Milstein scheme and the P-Milstein scheme for the system (34)

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Figure 8. Root mean-square convergence orders of the Milstein scheme and the P-Milstein scheme

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Figure 9. Evolution of $ H^0(y) $ and $ H^1(y) $ by the Milstein scheme and the P-Milstein scheme for system (35)

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Figure 10. Evolution of $ y^2 $ by the Milstein scheme and the P-Milstein scheme for the system (35)

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