Quasi-symplectic methods for Langevin-type equations

Langevin type equations are an important and fairly large class of systems close to Hamiltonian ones. The constructed mean-square and weak quasi-symplectic methods for such systems degenerate to symplectic methods when a system degenerates to a stochastic Hamiltonian one. In addition, quasi-symplectic methods' law of phase volume contractivity is close to the exact law. The methods derived are based on symplectic schemes for stochastic Hamiltonian systems. Mean-square symplectic methods were obtained in Milstein et al. (2002, SIAM J Numer Anal., 39, 2066-2088; 2003, SIAM J Numer Anal., 40, 1583-1604) while symplectic methods in the weak sense are constructed in this paper. Special attention is paid to Hamiltonian systems with separable Hamiltonians and with additive noise. Some numerical tests of both symplectic and quasisymplectic methods are presented. They demonstrate superiority of the proposed methods in comparison with standard ones.