Robust Lagrangian numerical schemes in computing effective diffusivities for chaotic and random flows

Abstract

Effective diffusivities of passive scalars diffusion in incompressible velocity fields have theoretical and practical importance. In this thesis, efforts have been made to develop a Lagrangian approach to calculate effective diffusivities and to analyze the error and physical phenomena based on numerical results.

Our approach is to integrate the stochastic differential equations of the particles by the proposed discrete schemes via Monte Carlo methods. To compute the effective diffusivities, we take the variance of the sampled positions divided by computational time. The computational time should be longer than the mixing time of dynamics, so the discrete schemes should preserve the inherent structures of the dynamics.

Via backward error analysis techniques, we proved the proposed schemes converge asymptotically with respect to the time step. Later on, we developed a new proof to show the convergence is uniform in computational time. The key ingredient of the proof is to propose discrete type cell problems, which are analogous to cell problems in traditional parabolic homogenization theory. And we concluded the schemes should preserve the invariant measure on torus space introduced by the periodicity of velocity fields. We generalized the proof to time-dependent cases and random cases.

Numerical examples were presented to verify the convergence in each case. We calculated the effective diffusivities of chaotic and random flows, including the Taylor Green field in two dimensions, the Arnold-Beltrami-Childress flow and Kolmogorov flow in three dimensions and also their generalizations to time-dependent and random cases. We investigated the convection-enhanced diffusion phenomenon in the large Péclet number regime. Our results showed that the diffusion enhancement has a strong correlation to mixing time and Lyapunov exponent.