Modified equations for weakly convergent stochastic symplectic schemes via their generating functions (Q329031)

Based on the underlying generating functions, the authors construct modified equations of weak \(k+k'\) order (where \(k'\in{\mathbb N}\)), which are beyond \(k\)th-order weakly convergenct stochastic symplectic methods. Their approach works for stochastic Hamiltonian systems with additive noises and, quoting from the abstract, with ``multiplicative noises but for which the Hamiltonian functions \(H_r(p,q)\), \(r\geq 1\), associated to the diffusion parts depend only on \(p\) or on \(q\)''. In these situations, it is shown that the modified equations of the weakly convergent stochastic symplectic methods are perturbed stochastic Hamiltonian systems of the original equations. Several numerical tests illustrate the approach.