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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Root mean-square convergence orders of the Milstein scheme, the Klöden scheme, the P-Milstein scheme, and the P-Klöden scheme
Evolution of
Evolution of
Evolution of the Casimir function by the Milstein scheme and the P-Milstein scheme
Root mean-square convergence orders of the Euler scheme, the Milstein scheme, the P-Euler scheme, and the P-Milstein scheme
Evolution of
A sample path of
Root mean-square convergence orders of the Milstein scheme and the P-Milstein scheme
Evolution of
Evolution of