Computing Effective Diffusivity of Chaotic and Stochastic Flows Using Structure-Preserving Schemes

Abstract

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.

Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions

“…For SDEs driven by standard Brownian motion, there are several types of weak stochastic modified equations with many applications, such as constructing high weak order numerical schemes, studying invariant measures of numerical schemes and investigating the mathematical mechanism of stochastic symplectic methods for stochastic Hamiltonian systems. We refer to [1,2,9,26,27,35,36,37] for interested readers. As to numerical schemes constructed by a thirdorder Taylor expansion for SDEs driven by fBm, the remainder term of the local error between Y and Y n has higher regularity, and the optimal strong convergence rate is obtained in [3] for H ∈ ( 1 4 , 1 2 ).…”

Convergence analysis of a Lagrangian numerical scheme in computing effective diffusivity of 3D time-dependent flows

“…In [32], we proposed a stochastic structure-preserving scheme based on a Lie-Trotter splitting scheme to solve the SDE (9). Specifically, we split the problem (9) into a deterministic subproblem,…”

“…Theorem 4.7. Let X n = (x n 1 , x n 2 ) T denote the solution of the two-dimensional passive tracer model (9) obtained by using our numerical scheme (32) with time step ∆t. If the Hamiltonian function H(t, x 1 , x 2 ) is separable, periodic and smooth (in order to guarantee the existence and uniqueness of the solution to the SDE (9)), then we can prove that the second-order moment of the solution X n (which can be viewed as a discrete Markov process) is at most linear growth, i.e.,…”

“…Recently, we developed a robust structure-preserving Lagrangian scheme to compute the effective diffusivity for chaotic and stochastic flows in [32]. We also obtained a rigorous error estimate for the numerical scheme in [32]. Specifically, let D E,num denote the numerical effective diffusivity obtained by our method.…”

A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model

“…A number of efficient numerical methods have been developed to solve stochastic PDEs, such as polynomial chaos [69,116,117], stochastic Galerkin method [8,35,86,98], stochastic collocation method [7,59], sparse grid methods [11,87,90,91], multilevel Monte Carlo method [14,29,42,52,73,97,104], and many others [9,12,27,82,103,107,109,112,114,115,118,121,122]. These methods have also been applied to solve the stochastic optimization and control problems [4,13,34,58,105].…”