Regularized Fluctuating Lattice Boltzmann Model

TL;DR

The paper tackles the integration of thermal fluctuations into lattice Boltzmann simulations while preserving fluctuation–dissipation balance for both hydrodynamic and ghost modes. It develops a regularized fluctuating lattice Boltzmann method (Reg-FLBM) on the $D3Q27$ lattice by projecting onto a complete Hermite basis and applying recursive regularization with distinct relaxation rates for hydrodynamic and ghost modes, plus noise terms that satisfy the fluctuation–dissipation theorem. Numerical tests demonstrate that Reg-FLBM reproduces correct fluctuation amplitudes across a wide range of viscosities, outperforming the conventional BGK-FLBM in stability and accuracy, and spectral analyses confirm fidelity of the fluctuation spectra. The method scales efficiently on multi-GPU HPC platforms, enabling large-scale studies of fluctuation-driven phenomena in mesoscale and nanoscale fluids, with promising avenues including higher-order lattices, non-ideal/multi-component extensions, and improved Galilean-invariance handling.

Abstract

We introduce a regularized fluctuating lattice Boltzmann model (Reg-FLBM) for the D3Q27 lattice, which incorporates thermal fluctuations through Hermite-based projections to ensure compliance with the fluctuation-dissipation theorem. By leveraging the recursive regularization framework, the model achieves thermodynamic consistency for both hydrodynamic and ghost modes. Compared to the conventional single-relaxation-time BGK-FLBM, the Reg-FLBM provides improved stability and a more accurate description of thermal fluctuations. The implementation is optimized for large-scale parallel simulations on GPU-accelerated architectures, enabling systematic investigation of fluctuation-driven phenomena in mesoscale and nanoscale fluid systems.

Figures (3)

• Figure 1: Equilibration ratio spectra for the four main hydrodynamic fields, computed as a function of the wavevector modulus $k$, for both the BGK-FLBM (top row) and the Reg-FLBM (bottom row), at two values of the relaxation time: $\tau=0.51$ (left column) and $\tau=50.0$ (right column). Panels: (a) BGK-FLBM, $\tau=0.51$; (b) BGK-FLBM, $\tau=50.0$; (c) Reg-FLBM, $\tau=0.51$; (d) Reg-FLBM, $\tau=50.0$.
• Figure 2: Minimum and maximum values of the equilibration ratio for the total momentum, $\mathrm{ER}\left(\sum_\alpha \rho u_\alpha\right)$, as a function of the relaxation time $\tau$, for the Reg-FLBM (blue, circles and squares) and the BGK-FLBM (orange, triangles and diamonds). For each value of $\tau$, the minimum and maximum deviation from unity are computed over all wavevector magnitudes $|\mathbf{k}|$ in the range $4 \leq |\mathbf{k}| \leq 64$.
• Figure 3: Weak scaling of GLUPS (Giga Lattice Updates Per Second) as a function of the number of GPUs, $N_{\mathrm{GPU}}$, for the regularized fluctuating (Reg-FLBM) and regularized non-fluctuating (Reg-LBM) models on Leonardo at CINECA. Each GPU handles a $256^3$ subdomain. Dashed and dotted lines show ideal speedup. Both models achieve nearly ideal scaling up to 64 GPUs, with Reg-FLBM reflecting the additional cost of fluctuations.