Temporal integrators for fluctuating hydrodynamics

Steven Delong, Boyce E. Griffith, Eric Vanden-Eijnden, and Aleksandar Donev

Phys. Rev. E 87, 033302 – Published 11 March 2013

Abstract

Including the effect of thermal fluctuations in traditional computational fluid dynamics requires developing numerical techniques for solving the stochastic partial differential equations of fluctuating hydrodynamics. These Langevin equations possess a special fluctuation-dissipation structure that needs to be preserved by spatio-temporal discretizations in order for the computed solution to reproduce the correct long-time behavior. In particular, numerical solutions should approximate the Gibbs-Boltzmann equilibrium distribution, and ideally this will hold even for large time step sizes. We describe finite-volume spatial discretizations for the fluctuating Burgers and fluctuating incompressible Navier-Stokes equations that obey a discrete fluctuation-dissipation balance principle just like the continuum equations. We develop implicit-explicit predictor-corrector temporal integrators for the resulting stochastic method-of-lines discretization. These stochastic Runge-Kutta schemes treat diffusion implicitly and advection explicitly, are weakly second-order accurate for additive noise for small time steps, and give a good approximation to the equilibrium distribution even for very strong fluctuations. Numerical results demonstrate that a midpoint predictor-corrector scheme is very robust over a broad range of time step sizes.

Received 5 December 2012

DOI:https://doi.org/10.1103/PhysRevE.87.033302

Authors & Affiliations

Steven Delong¹, Boyce E. Griffith², Eric Vanden-Eijnden^1,*, and Aleksandar Donev^1,†

¹Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
²Leon H. Charney Division of Cardiology, Department of Medicine, New York University School of Medicine, New York, New York 10016, USA

^*eve2@courant.nyu.edu
^†donev@courant.nyu.edu

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Vol. 87, Iss. 3 — March 2013

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Images

Figure 1
Static structure factor $S_{κ}$ for a periodic system of 256 cells at thermodynamic equilibrium. Left panel: Midpoint scheme (29) with $w_{1} = 1 / 2$ and non-Hamiltonian advection (40) for time step size $β = 0.4$ for weak fluctuations $ε = 0.1$ ( $α \approx 0.13$ ), medium fluctuations $ε = 1$ ( $α = 0.4$ ), and strong fluctuations $ε = 4$ ( $α = 0.8$ ). For $ε = 4$ , reducing the time step to $β = 0.1$ ( $α = 0.2$ ) does not significantly improve the accuracy and even if $Δ t \to 0$ the wrong spectrum of fluctuations is obtained. Switching to the Hamiltonian discretization (12) significantly lowers the errors. Right panel: Comparison between the trapezoidal scheme (28) and the midpoint scheme (29) with $w_{1} = 1 / 2$ and with $w_{1} = 1 / 4$ , for $ε = 4$ and large time step size $β = 0.5$ ( $α = 1$ ) and small time step $β = 0.125$ ( $α = 0.25$ ). Advection is discretized in a Hamiltonian manner.Reuse & Permissions
Figure 2
Static structure factor $S_{κ}$ at thermodynamic equilibrium in the case of weaker fluctuations and large time step sizes, for the trapezoidal scheme (28) and for the midpoint scheme (29) with $w_{1} = 1 / 2$ and with $w_{1} = 1 / 4$ . Left panel: Moderate fluctuations, $ε = 0.1$ , for time step size $β = 10$ , $α \approx 3.2$ , $γ = 1$ . Right panel: Weak fluctuations, $ε = 0.01,$ and time time step size $β = 100$ , $α = 10$ , $γ = 1$ .Reuse & Permissions
Figure 3
Discrete structure factors $S_{κ}^{(Ω)}$ (top) and $S_{κ}^{(c)}$ (bottom) for the midpoint scheme (29) with $w_{1} = 1 / 2$ on the left and for the trapezoidal scheme (28) on the right, for the case of large fluctuations $ε = 4$ . Top row: Velocity spectra for a time step size $β = 0.5$ , $α = 1$ . Bottom row: concentration spectra for $β_{c} = χ β / ν = 0.0625$ and $α = 0.5$ .Reuse & Permissions
Figure 4
Average error in the static spectra for several time step sizes for the trapezoidal scheme (28) and for the midpoint scheme (29) with $w_{1} = 1 / 2$ , for the case of large fluctuations $ε = 4$ . First- and second-order error trends are indicated, showing the second-order asymptotic accuracy. Error bars are comparable to the symbol size and not shown. Left panel: $〈 S_{κ}^{(Ω)} - 1 〉$ . Right panel: $〈 S_{κ}^{(c)} - 1 〉$ .Reuse & Permissions
Figure 5
Discrete structure factor for concentration $S_{κ}^{(c)}$ for the midpoint scheme (29) with $w_{1} = 1 / 2$ on the left and for the trapezoidal scheme (28) on the right, for the case of weak fluctuations, $ε = 0.01$ , and large time step size $β = 50$ , $β_{c} = 12.5$ , $α = 5$ .Reuse & Permissions
Figure 6
Comparison of $\exp (- x)$ with three rational approximations. The approximation in the Crank-Nicolson scheme (23) does not decay to zero for large $x$ . The approximation (B1) decays to zero for $w_{4} = 1 \pm \sqrt{2} / 2$ ; however, only the positive sign gives a strictly positive approximation.Reuse & Permissions

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