Vol. 5, No. 2, 2010

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On the accuracy of finite-volume schemes for fluctuating hydrodynamics

Aleksandar Donev, Eric Vanden-Eijnden, Alejandro Garcia and John Bell

Vol. 5 (2010), No. 2, 149–197
Abstract

This paper describes the development and analysis of finite-volume methods for the Landau–Lifshitz Navier–Stokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white-noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. Originally derived for equilibrium fluctuations, the LLNS equations have also been shown to be accurate for nonequilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected nonequilibrium flows. In this paper, we introduce a more systematic approach based on studying discrete equilibrium structure factors for a broad class of explicit linear finite-volume schemes. This new approach provides a better characterization of the accuracy of a spatiotemporal discretization as a function of wavenumber and frequency, allowing us to distinguish between behavior at long wavelengths, where accuracy is a prime concern, and short wavelengths, where stability concerns are of greater importance. We use this analysis to develop a specialized third-order Runge–Kutta scheme that minimizes the temporal integration error in the discrete structure factor at long wavelengths for the one-dimensional linearized LLNS equations. Together with a novel method for discretizing the stochastic stress tensor in dimension larger than one, our improved temporal integrator yields a scheme for the three-dimensional equations that satisfies a discrete fluctuation-dissipation balance for small time steps and is also sufficiently accurate even for time steps close to the stability limit.

Keywords
finite-volume scheme, hydrodynamics
Mathematical Subject Classification 2000
Primary: 35K05, 65C30, 65N12, 65N40
Milestones
Received: 12 June 2009
Revised: 18 December 2009
Accepted: 22 April 2010
Published: 16 June 2010
Authors
Aleksandar Donev
Lawrence Berkeley National Laboratory
Center for Computational Sciences and Engineering
MS 50A-1148, LBL
1 Cyclotron Rd.
Berkeley 94720
United States
http://cims.nyu.edu/~donev
Eric Vanden-Eijnden
New York University
Courant Institute of Mathematical Sciences
New York 10012
United States
Alejandro Garcia
San Jose State University
Department of Physics and Astronomy
San Jose, CA 95192
United States
John Bell
Lawrence Berkeley National Laboratory
Center for Computational Science and Engineering
Berkeley, CA 94720
United States