Numerical methods for the stochastic Landau-Lifshitz Navier-Stokes equations

The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several computational fluid dynamics approaches are considered (including MacCormack’s two-step Lax-Wendroff scheme and the piecewise parabolic method) and are found to give good results for the variance of momentum fluctuations. However, neither of these schemes accurately reproduces the fluctuations in energy or density. We introduce a conservative centered scheme with a third-order Runge-Kutta temporal integrator that does accurately produce fluctuations in density, energy, and momentum. A variety of numerical tests, including the random walk of a standing shock wave, are considered and results from the stochastic LLNS solver are compared with theory, when available, and with molecular simulations using a direct simulation Monte Carlo algorithm.

Figure 1

(Color online) Spatial correlation of density fluctuations. Solid line is $⟨\delta {\rho }_i\delta {\rho }_j⟩=⟨\delta {\rho }^2⟩{\delta }_{i,j}^K$ [see Eqs. (A2, A3)].

Figure 2

(Color online) Spatial correlation of momentum fluctuations. Solid line is $⟨\delta J_i\delta J_j⟩=⟨\delta J^2⟩{\delta }_{i,j}^K$ [see Eqs. (A4, A6)].

Figure 3

(Color online) Spatial correlation of energy fluctuations. Solid line is $⟨\delta E_i\delta E_j⟩=⟨\delta E^2⟩{\delta }_{i,j}^K$ [see Eqs. (A5, A7)].

Figure 4

(Color online) Spatial correlation of density-momentum fluctuations.

Figure 5

(Color online) Time correlation of density fluctuations for equilibrium problem, on a periodic domain (left panel) and a domain with specular wall boundaries (right panel).

Figure 6

Spatial correlation of density and momentum fluctuations for a system subjected to a temperature gradient. Compare with Fig. 4.

Figure 7

(Color online) Variance of shock location for mass density profile (left panel, $⟨\delta {\sigma }_{\rho }{\left(t\right)}^2⟩$) and pressure profile (right panel, $⟨\delta {\sigma }_P{\left(t\right)}^2⟩$). Estimated variances (4000-run ensembles) vs time $t$ for a deterministically steady shock of Mach number 1.2, 1.4, or 2.0. Solid lines are for RK3, dashed lines are from DSMC molecular simulations.