[Paper Review] On the Accuracy of Finite-Volume Schemes for Fluctuating Hydrodynamics

A. Donev, Eric Vanden‐Eijnden+2 more|ArXiv.org|Jun 12, 2009

Fluid Dynamics and Turbulent Flows53 references4 citations

TL;DR

This paper develops a high-accuracy finite-volume scheme for the Landau-Lifshitz Navier-Stokes (LLNS) equations by introducing a wavenumber- and frequency-dependent analysis of discrete equilibrium structure factors. It presents a specialized third-order Runge-Kutta temporal integrator and a novel stochastic stress discretization, achieving accurate fluctuation-dissipation balance and minimal long-wavelength temporal errors in 3D simulations, even near stability limits.

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 non-equilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected non-equilibrium 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 spatio-temporal 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.

Motivation & Objective

• To systematically evaluate the accuracy of finite-volume schemes for the LLNS equations beyond equilibrium variance and correlation measurements.
• To address the challenge of balancing accuracy at long wavelengths with stability at short wavelengths in stochastic hydrodynamics simulations.
• To develop a temporal integrator that minimizes long-wavelength temporal errors in the discrete structure factor for the 1D linearized LLNS equations.
• To design a novel 3D discretization of the stochastic stress tensor that preserves discrete fluctuation-dissipation balance for small time steps.
• To construct a fully consistent 3D scheme that maintains accuracy even when time steps approach the stability limit.

Proposed method

• Uses discrete equilibrium structure factors as a diagnostic to assess spatio-temporal discretization accuracy across wavenumbers and frequencies.
• Derives a specialized third-order Runge-Kutta scheme optimized to minimize temporal integration error in the structure factor at long wavelengths for 1D linearized LLNS.
• Introduces a new method for discretizing the stochastic stress tensor in dimensions greater than one, ensuring consistent fluctuation-dissipation balance.
• Applies a Fourier-space analysis to decouple wavevector modes and evaluate discrete structure factors under various schemes.
• Combines the optimized temporal integrator with the new stress discretization to construct a 3D finite-volume scheme with improved accuracy and stability.
• Validates the scheme by demonstrating exact discrete fluctuation-dissipation balance for small time steps and robust accuracy near the stability limit.

Experimental results

Research questions

• RQ1 How can the accuracy of finite-volume schemes for the LLNS equations be systematically quantified across different wavenumbers and frequencies?
• RQ2 What temporal integration scheme minimizes long-wavelength temporal errors in the discrete structure factor for the 1D linearized LLNS equations?
• RQ3 How can the stochastic stress tensor be discretized in 3D to preserve the discrete fluctuation-dissipation relation?
• RQ4 To what extent can the time step be increased while maintaining accurate fluctuation-dissipation balance and structure factor accuracy?
• RQ5 Can a 3D finite-volume scheme be constructed that is both accurate at long wavelengths and stable near the time step limit?

Key findings

• The proposed third-order Runge-Kutta scheme minimizes temporal integration error in the discrete structure factor at long wavelengths for the 1D linearized LLNS equations.
• The new 3D stochastic stress discretization ensures that the scheme satisfies the discrete fluctuation-dissipation relation for small time steps.
• The combined scheme maintains accurate structure factors and fluctuation-dissipation balance even when time steps are close to the stability limit.
• The Crank-Nicolson semi-implicit method achieves a perfect static structure factor (S_k = 1) independent of time step, but suffers from spatial discretization errors.
• For advection-diffusion systems, the semi-implicit scheme with predictor-corrector treatment preserves S_k = 1 when r = 0 and matches leading-order error of PC-2RNG when |r| > 0.
• The analysis via structure factors provides a superior diagnostic for distinguishing long-wavelength accuracy from short-wavelength stability concerns.