Acceleration Methods for Coarse-Grained Numerical Solution of the Boltzmann Equation

[+] Author and Article Information
Husain A. Al-Mohssen, Nicolas G. Hadjiconstantinou

Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA 02139

Ioannis G. Kevrekidis

Department of Chemical Engineering and PACM, Princeton University, Princeton, NJ 08540

Although implicit solution methods based on a finite difference discretization of the Boltzmann equation have also been developed, these have been found to converge slowly or not at all for lower values of Kn 15.

E.g., when the initial Jacobian corresponds to a different flow profile.

The Knudsen layer is “extracted” by subtracting the straight-line (slip-flow) Navier-Stokes result 21.

J. Fluids Eng 129(7), 908-912 (Dec 04, 2006) (5 pages) doi:10.1115/1.2742725 History: Received June 01, 2006; Revised December 04, 2006

We present a coarse-grained steady-state solution framework for the Boltzmann kinetic equation based on a Newton-Broyden iteration. This approach is an extension of the equation-free framework proposed by Kevrekidis and coworkers, whose objective is the use of fine-scale simulation tools to directly extract coarse-grained, macroscopic information. Our current objective is the development of efficient simulation tools for modeling complex micro- and nanoscale flows. The iterative method proposed and used here consists of a short Boltzmann transient evolution step and a Newton-Broyden contraction mapping step based on the Boltzmann solution; the latter step only solves for the macroscopic field of interest (e.g., flow velocity). The predicted macroscopic field is then used as an initial condition for the Boltzmann solver for the next iteration. We have validated this approach for isothermal, one-dimensional flows in the low Knudsen number regime. We find that the Newton-Broyden iteration converges in O(10) iterations, starting from arbitrary guess solutions and a Navier-Stokes based initial Jacobian. This results in computational savings compared to time-explicit integration to steady states when the time to steady state is longer than O(40) mean collision times.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 4

Exact and Broyden Knudsen layer

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Figure 5

Convergence history for a Kn=0.1 problem. The error is averaged over all spatial nodes (512 Total) and an L-1 Norm is used for the error.

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Figure 3

Flowchart of the solution algorithm as explained in the text

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Figure 2

Sketch of different steps of the maturing procedure

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Figure 1

Sketch of 1D computational domain




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