Symplectic methods based on Pad$\acute{e}$ approximation for some stochastic Hamiltonian systems
This work provides new symplectic integrators for stochastic Hamiltonian systems, which are important for long-time simulations in physics and engineering, but the results are incremental as they extend deterministic Padé methods to specific stochastic cases.
The paper introduces numerical schemes based on Padé approximation for stochastic Hamiltonian systems, achieving symplecticity and mean-square orders up to (r+s)/2 for linear systems and up to r̂+ŝ+1 for additive noise systems.
In this article, we introduce a kind of numerical schemes, based on Pad$\acute{e}$ approximation, for two stochastic Hamiltonian systems which are treated separately. For the linear stochastic Hamiltonian systems, it is shown that the applied Pad$\acute e$ approximations $P_{(k,k)}$ give numerical solutions that inherit the symplecticity and the proposed numerical schemes based on $P_{(r,s)}$ are of mean-square order $\frac{r+s}{2}$ under appropriate conditions. In case of the special stochastic Hamiltonian systems with additive noises, the numerical method using two kinds of Pad$\acute e$ approximation $P_{(\hat r,\hat s)}$ and $P_{(\check r,\check s)}$ has mean-square order $\check r+\check s+1$ when $\hat r+\hat s=\check r+\check s+2$. Moreover, the numerical solution is symplectic if $\hat r=\hat s$.