Effective diffusivities in volume preserving (incompressible) chaotic and stochastic flows are well-characterized based on the analysis of corrector problems in homogenization theory. However, the corrector problems (stochastic partial differential equations in general) are highly expensive to compute in the advection-dominated regime especially in three space dimensions due to the presence of sharp internal layers. We develop Lagrangian approximation theory based on structure-preserving discretization (a.k.a. symplectic integrator) of stochastic differential equations (SDEs). We identify a discrete corrector problem in terms of the SDE evolution and show its convergence to the PDE corrector (Eulerian) problem. The growth rate of the second moment of SDE solutions (Lagrangian trajectories) approximates the exact effective diffusivity uniformly in time. The Lagrangian approach is grid-free and parallelizable. Computational results on Arnold-Beltrami-Childress (ABC) and Kolmogorov flows, space-time stochastic flows, validate our theory and help explore the scaling behavior of effective diffusivity in the zero molecular diffusivity limit. The fronts speeds of Kolmogorov-Petrovsky-Piskunov equations are effective Hamiltonians of quadratically nonlinear Hamilton-Jacobi equations. In similar spirit, Lagrangian approximation based on Feynman-Kac formula and operator splitting is developed with encouraging computations on chaotic flows. This is joint work with J. Lyu, Z. Wang, Z. Zhang at Hong Kong University.
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