Decomposing the collision operator in the lattice Boltzmann method

TL;DR

The paper introduces the Composite Collision Framework (CCF) for the lattice Boltzmann method, enabling any collision operator to be decomposed into a convex sum of simpler component collisions: $\Omega[f_i] = \sum_n \eta^n \Omega^n[f_i]$ with $\sum_n \eta^n = 1$. This formalism allows analyzing and constructing collision steps via component populations $f_i^n$, densities $\rho^n$, and momenta $(\rho\mathbf{u})^n$, including half-source corrections, so that $\rho = \sum_n \rho^n$ and $\rho\mathbf{u} = \sum_n (\rho\mathbf{u})^n$. The framework is used to reinterpret Robin boundary conditions as compositions of Bounce-Back (BB) and Anti-Bounce-Back (ABB) steps, and to develop new composite collisions such as Partial RBC (PRBC) and Partial Bounceback with Fluxes (PBBF), while detailing force incorporation within CCF. Overall, CCF provides a modular, interpretable pathway to model complex transport phenomena, including reactive membranes and porous media, with potential for improved design and analysis of lattice Boltzmann simulations.

Abstract

In transport theory, physical phenomena are well described using the Boltzmann equation, which is efficiently simulated and discretized with the lattice Boltzmann method. The collision step defines the microscopic molecules behavior, and thus the simulated physical phenomena. For complex phenomena, the collision step becomes complex as well. In this paper, we propose a framework to systematically decompose the collision step into individual collision rules. Each collision rule is easier to understand, thus a faster understanding of the whole is achieved. By inverting the process, i.e. composing multiple collision rules together, one can create novel collision steps, which can better describe the underlying complex phenomena. This framework's applications are manyfold, from both a theoretical and an application standpoint. Shown here is the decomposition of Robin boundary condition into Dirichlet and Neumann boundary conditions, extending it to a partial Robin boundary condition, and semi-permeable reactive membranes.

Figures (5)

• Figure 1: Steady-state simulation results comparing the novel PRBC to resolved BB-RBC setup. A spacing of $N_\mathrm{BB}=5$ with a $\mathrm{Da}=500$ is used. For the PRBC, the composite fraction used is $\eta^\mathrm{RBC}=1/9$. The inset shows the behavior near the wall to highlight the effects in that region.
• Figure 2: Comparison of non-dimensional reaction rates from boundary condition using the alternating (pluses) vs PRBC (circles). The spacing $N_\mathrm{BB}$ is varied between 1 and 30. The Damköhler numbers $\mathrm{Da}=\{0.5, 5, 50, 500, 5000\}$ were simulated, where the fitting parameter $A=\{1, 1, 1.3, 1.9, 2.0\}$ were used, respectively.
• Figure 3: Simulation results of semi-permeable diffusion with a reactive membrane in between the dashed lines. The impact of variations of the reaction rate is shown. The black line is the common baseline. The composite fraction $\eta^\mathrm{RBC}=0.1$.
• Figure 4: Simulation results of semi-permeable diffusion with a reactive membrane in between the dashed lines. The impact of variations of the composite fractions $\eta^\mathrm{RBC}$ are shown. The black line is the common baseline. The reaction rate is constant at $\mathrm{Da}=0.08$.
• Figure B.1: Sensitivity of the fitting parameter $A$ on simulated non-dimensional reaction rated. The reaction rates of the boundary condition using the alternating BC (black pluses) vs PRBC (colored circles) are shown. The spacing $N_\mathrm{BB}$ is varied between 1 and 30. The Damköhler number $\mathrm{Da}=50$ was simulated with fitting parameters $A=\{1,\ 1.1,\ 1.3,\ 1.5\}$.