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 $q_{n+1}=q_n+\frac{1}{2}(p_n+p_{n+1})\Delta t$, whose mean-square velocity order increases with the number of stages.