TY - JOUR

T1 - On the accuracy of finite-volume schemes for fluctuating hydrodynamics

AU - Donev, Aleksandar

AU - Vanden-Eijnden, Eric

AU - Garcia, Alejandro

AU - Bell, John

N1 - Funding Information:
MSC2000: 35K05, 65C30, 65N12, 65N40. Keywords: finite-volume scheme, hydrodynamics. Donev’s work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. The work of Bell and Garcia was supported by the Applied Mathematics Research Program of the U. S. Department of Energy under contract no. DE-AC02-05CH11231. The work of Vanden-Eijnden was supported by the National Science Foundation through grants DMS02-09959, DMS02-39625, and DMS07-08140, and by the Office of Naval Research through grant N00014-04-1-0565.
Funding Information:
The authors thank Berni Alder and Jonathan Goodman for helpful discussions, and Paul Atzberger for inspiring perspectives on the discrete fluctuation dissipation relation and a critical reading of this paper. Donev's work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. The work of Bell and Garcia was supported by the Applied Mathematics Research Program of the U. S. Department of Energy under contract no. DE-AC02-05CH11231. The work of Vanden-Eijnden was supported by the National Science Foundation through grants DMS02-09959, DMS02-39625, and DMS07-08140, and by the Office of Naval Research through grant N00014-04-1-0565.

PY - 2010

Y1 - 2010

N2 - 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 whitenoise 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.

AB - 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 whitenoise 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.

KW - Finite-volume scheme

KW - Hydrodynamics

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U2 - 10.2140/camcos.2010.5.149

DO - 10.2140/camcos.2010.5.149

M3 - Article

AN - SCOPUS:85016139985

SN - 1559-3940

VL - 5

SP - 149

EP - 197

JO - Communications in Applied Mathematics and Computational Science

JF - Communications in Applied Mathematics and Computational Science

IS - 2

ER -