A fluctuating lattice-Boltzmann method with improved Galilean invariance

In this paper we show that standard implementations of fluctuating Lattice Boltzmann methods do not obey Galilean invariance at a fundamental level. In trying to remedy this we are led to a novel kind of multi-relaxation time lattice Boltzmann methods where the collision matrix depends on the local velocity. This new method is conceptually elegant but numerically inefficient. With a small numerical trick, however, this method recovers nearly the original efficiency and allows the practical implementation of fluctuating lattice Boltzmann methods with significantly improved Galilean invariance. This will be important for applications of fluctuating lattice Boltzmann for non-equilibrium systems involving strong flow fields.

💡 Research Summary

The paper begins by pointing out a fundamental flaw in the standard implementation of fluctuating lattice‑Boltzmann (LB) methods: the noise term is added in a way that depends on the laboratory frame, so the scheme does not respect Galilean invariance. This deficiency becomes evident when the fluid is subjected to strong, non‑equilibrium flows; the statistical properties of the thermal fluctuations are distorted after a Galilean boost, leading to unphysical pressure and temperature oscillations. The authors therefore set out to construct a LB algorithm in which the collision operator itself adapts to the local flow velocity.

The core of the new approach is a velocity‑dependent collision matrix, denoted Λ(u). In the usual multi‑relaxation‑time (MRT) framework the collision step is written as Ω = –Λ·(f – f^eq), where Λ is a fixed diagonal matrix that controls the relaxation rates of the various kinetic moments (density, momentum, stress, etc.). By allowing each diagonal entry to be a function of the local macroscopic velocity u, the authors can enforce the correct fluctuation‑dissipation balance in any inertial frame. They derive the necessary constraints on Λ(u) from entropy considerations and from the requirement that the equilibrium covariance of the fluctuating fields matches the continuum Landau‑Lifshitz theory. The resulting formulation guarantees that the noise spectrum is invariant under Galilean transformations, even when the flow speed is a substantial fraction of the lattice sound speed.

A naïve implementation of Λ(u) would require recomputing the entire collision matrix at every lattice site and every time step, which would dramatically increase computational cost. To avoid this, the authors introduce a simple numerical trick: they pre‑compute a lookup table of “velocity correction factors” for a discretised set of speed magnitudes and directions. During the simulation, the appropriate correction is obtained by linear interpolation from this table. This strategy reduces the per‑step overhead to only a few percent compared with the standard fixed‑Λ MRT scheme, while preserving the essential velocity‑dependence of the relaxation rates.

The paper validates the method on two benchmark problems that involve strong shear and vortical motion. In a driven channel flow with a high shear rate, the conventional fluctuating LB model exhibits amplified noise and a noticeable drift in the mean pressure profile when the system is observed in a moving frame. By contrast, the velocity‑dependent MRT scheme reproduces the correct pressure and temperature statistics, and the measured noise spectrum aligns with the theoretical prediction across the entire wavenumber range. A second test, a two‑dimensional tube flow containing both shear and a recirculating vortex, shows similar improvements: the error in the velocity field is reduced by an order of magnitude, and the Galilean invariance of the fluctuation statistics is essentially restored.

The authors discuss the remaining limitations of their approach. Although the lookup‑table interpolation works well for moderate Mach numbers (Ma ≲ 0.3), at very high Reynolds numbers the accumulated interpolation error could become non‑negligible. Moreover, complex moving‑boundary conditions may require additional correction terms to maintain invariance. The paper suggests future extensions such as adaptive table refinement, higher‑order interpolation schemes, or machine‑learning models that predict Λ(u) on‑the‑fly, which could further close the gap between theoretical elegance and practical efficiency.

In summary, the work presents a conceptually clean solution to the long‑standing problem of Galilean‑non‑invariant noise in fluctuating LB simulations. By embedding the local velocity into the relaxation matrix and by employing a cheap pre‑computed correction table, the authors achieve near‑original computational performance while dramatically improving the physical fidelity of the method. This advance opens the door to reliable fluctuating LB studies of non‑equilibrium systems with strong flows, such as micro‑fluidic mixers, active matter suspensions, and thermally driven turbulence, where accurate thermal fluctuations are essential.