Effective particle methods in calculating effective diffusivities and KPP type front speeds in periodic, chaotic and random flow fields

Abstract

Passive scalars diffusion in incompressible velocity fields is of great theoretical and practical importance, and front propagation is also a fundamental problem in many scientific areas. In this thesis, efforts have been made to develop a Lagrangian approach to calculate effective diffusivities and KPP type front speeds in periodic, chaotic, or random flow fields and to analyze the error and physic phenomenons based on numerical results.

Traditional methods like Finite Element Methods, Fourier Spectral Methods suffer a lot in the curse of dimensions and work badly in large P$\acute{\text{e}}$clet number regime when solving these problems. In our approach, integrating the stochastic differential equations of particles by proposed discrete schemes via Monte Carlo methods can handle these problems.

To compute the effective diffusivities in random flows, we propose stochastic structure-preserving schemes to solve the SDEs. The convergence analysis follows a probabilistic approach, which interprets the solution process generated by our numerical schemes as a Markov process. By exploring the ergodicity of the solution process, we obtain a convergence analysis of our method in computing long-time solutions of the SDEs. Most importantly, our analysis result reveals the equivalence of the definition of effective diffusivity by solving discrete-type and continuous-type corrector problems, which is fundamental and interesting. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method and investigate the convection-enhanced diffusion phenomenon in two- and three-dimensional incompressible random flows.

To compute the KPP type front speeds, the variational principle reduces the computation of KPP front speeds to a principal eigenvalue problem on a periodic domain of a linear advection-diffusion operator with space-time periodic coefficients. We develop efficient Lagrangian methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting methods for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical methods. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress (ABC) flow and time-dependent Kolmogorov flow in three-dimensional space.