ACCURATE STATIONARY DENSITIES WITH PARTITIONED NUMERICAL METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS

We devise explicit partitioned numerical methods for second-order-in-time scalar stochastic differential equations, using one Gaussian random variable per timestep. The construction proceeds by analysis of the stationary density in the case of constant-coefficient linear equations, imposing exact stationary statistics in the position variable and absence of correlation between position and velocity; the remaining error is in the velocity variable. A new two-stage reverse leapfrog method has good properties in the position variable and is symplectic in the limit of zero damping. Explicit new Runge-Kutta leapfrog methods are constructed, sharing the property that q(n+1) = q(n) + 1/2 (p(n) + p(n+1))Delta t, whose mean-square velocity order increases with the number of stages.