  In this paper we study the problem of computing the effective diffusivity for
a particle moving in chaotic and stochastic flows. In addition we numerically
investigate the residual diffusion phenomenon in chaotic advection. The
residual diffusion refers to the non-zero effective (homogenized) diffusion in
the limit of zero molecular diffusion as a result of chaotic mixing of the
streamlines. In this limit traditional numerical methods typically fail since
the solutions of the advection-diffusion equation develop sharp gradients.
Instead of solving the Fokker-Planck equation in the Eulerian formulation, we
compute the motion of particles in the Lagrangian formulation, which is
modelled by stochastic differential equations (SDEs). We propose a new
numerical integrator based on a stochastic splitting method to solve the
corresponding SDEs in which the deterministic subproblem is symplectic
preserving while the random subproblem can be viewed as a perturbation. We
provide rigorous error analysis for the new numerical integrator using the
backward error analysis technique and show that our method outperforms standard
Euler-based integrators. Numerical results are presented to demonstrate the
accuracy and efficiency of the proposed method for several typical chaotic and
stochastic flow problems of physical interests.
