Michael V. Tretyakov

Ruslan L. Davidchack, Thomas E. Ouldridge, Michael V. Tretyakov

We introduce new Langevin-type equations describing the rotational and translational motion of rigid bodies interacting through conservative and non-conservative forces, and hydrodynamic coupling. In the absence of non-conservative forces the Langevin-type equations sample from the canonical ensemble. The rotational degrees of freedom are described using quaternions, the lengths of which are exactly preserved by the stochastic dynamics. For the proposed Langevin-type equations, we construct a weak 2nd order geometric integrator which preserves the main geometric features of the continuous dynamics. The integrator uses Verlet-type splitting for the deterministic part of Langevin equations appropriately combined with an exactly integrated Ornstein-Uhlenbeck process. Numerical experiments are presented to illustrate both the new Langevin model and the numerical method for it, as well as to demonstrate how inertia and the coupling of rotational and translational motion can introduce qualitatively distinct behaviours.

Zhongqiang Zhang, Michael V. Tretyakov, Boris Rozovskii et al.

We compare Wiener chaos and stochastic collocation methods for linear advection-reaction-diffusion equations with multiplicative white noise. Both methods are constructed based on a recursive multi-stage algorithm for long-time integration. We derive error estimates for both methods and compare their numerical performance. Numerical results confirm that the recursive multi-stage stochastic collocation method is of order $Δ$ (time step size) in the second-order moments while the recursive multi-stage Wiener chaos method is of order $Δ^{\mathsf{N}}+Δ^2$ ($\mathsf{N}$ is the order of Wiener chaos) for advection-diffusion-reaction equations with commutative noises, in agreement with the theoretical error estimates. However, for non-commutative noises, both methods are of order one in the second-order moments.