Stochastic symplectic partitioned Runge–Kutta methods for stochastic Hamiltonian systems with multiplicative noise

General order conditions for stochastic partitioned Runge–Kutta methods

“…It is similar to the nonpartitioned case, which for ODEs is discussed by Sanz-Serna and Abia [27] and generalized to SDEs in Anmarkrud and Kvaernø [3]. The conditions for preserving quadratic invariants for non-separable equations with two partitionings have been developed by Hong, Xu and Wang [19], see also Ma and Ding [19], both in the context of symplectic methods. Using similar ideas, it is possible to find conditions for a more general partitioning.…”

“…• All trees for which a node of shape q (q ∈ {1, 2}) is followed by a node of the same shape, Consider the following method, proposed in [23]: (19) (W m (s) − W m (t n ))ds. It is straightforward to show that φ(τ ) = Φ(τ ) for trees τ 1 , τ 2 , τ 7 , τ 8 in (18).…”

Issues in the Software Implementation of Stochastic Numerical Runge–Kutta

“…В виду крайней сложности дальнейшего повышения порядка точности стохастических численных схем, современные работы в основном концентрируются на получении численных схем для частных случаев СДУ. Можно выделить работы посвященные симплектическим стохастическим численным методам Рунге-Кутта [33][34][35] и неявным схемам, в частности стохастическим аналогам метода Розенброка [36].…”

Stochastic multi-symplectic Runge–Kutta methods for stochastic Hamiltonian PDEs

“…Especially Runge-Kutta (RK) methods are widely used. In the last decade, it has been widely recognized that the stochastic RK methods play an important role in numerically solving stochastic ODEs and popularly employed, see, e.g., [1,2,12,17,19,20,22,23,24] and the references therein. The symplectic condition of stochastic RK methods for stochastic Hamiltonian ODEs was obtained firstly in [19].…”

“…To the best of our knowledge, there has been no work in the literature which studies the multi-symplectic RK methods for stochastic Hamiltonian PDEs (1.1). Motivated by [7,19,20], we consider the general case of stochastic Hamiltonian PDEs, investigate the multi-symplecticity of stochastic RK methods, and present some sufficient conditions for stochastic multi-symplectic RK methods in this paper. To this end, we main utilize the corresponding variation form of stochastic Hamiltonian PDEs and the chain rule of Stratonovich integral, then derive the multi-symplectic conditions by a tedious computation.…”