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In this paper we study the problem of computing the effective diffusivity fora particle moving in chaotic and stochastic flows. In addition we numericallyinvestigate the residual diffusion phenomenon in chaotic advection. Theresidual diffusion refers to the non-zero effective (homogenized) diffusion inthe limit of zero molecular diffusion as a result of chaotic mixing of thestreamlines. In this limit traditional numerical methods typically fail sincethe solutions of the advection-diffusion equation develop sharp gradients.Instead of solving the Fokker-Planck equation in the Eulerian formulation, wecompute the motion of particles in the Lagrangian formulation, which ismodelled by stochastic differential equations (SDEs). We propose a newnumerical integrator based on a stochastic splitting method to solve thecorresponding SDEs in which the deterministic subproblem is symplecticpreserving while the random subproblem can be viewed as a perturbation. Weprovide rigorous error analysis for the new numerical integrator using thebackward error analysis technique and show that our method outperforms standardEuler-based integrators. Numerical results are presented to demonstrate theaccuracy and efficiency of the proposed method for several typical chaotic andstochastic flow problems of physical interests.

Computing effective diffusivity of chaotic and stochastic flows using structure preserving schemes