Parametric Lattice Boltzmann Method

The discretized equilibrium distributions of the lattice Boltzmann method are presented by using the coefficients of the Lagrange interpolating polynomials that pass through the points related to discrete velocities and using moments of the Maxwell-Boltzmann distribution. The ranges of flow velocity and temperature providing positive valued distributions vary with regulating discrete velocities as parameters. New isothermal and thermal compressible models are proposed for flows of the level of the isothermal and thermal compressible Navier-Stokes equations. Thermal compressible shock tube flows are simulated by only five on-lattice discrete velocities. Two-dimensional isothermal and thermal vortices provoked by the Kelvin-Helmholtz instability are simulated by the parametric models.

💡 Research Summary

The paper introduces a novel formulation of the equilibrium distribution in the lattice Boltzmann method (LBM) by exploiting the coefficients of Lagrange interpolating polynomials that pass through points defined by the discrete velocities and Kronecker deltas. By enforcing that the discrete moments of the redistribution rule match the moments of the continuous Maxwell‑Boltzmann distribution, the authors derive a general expression for the equilibrium weights r_i as a linear combination of the first q moments, where q is the number of discrete velocities. This construction guarantees that the first q − 1 moments are reproduced exactly, establishing a direct link between the number of velocities and the order of accuracy n* = q − 1.

Two classes of models are presented. For three velocities (v₁ = 0, v₂,₃ = ±√ζ θ₀) the parameter ζ controls the admissible flow speed range for which all r_i remain non‑negative. When ζ = 3 the model reduces to the classic LBGK scheme; ζ = 4 expands the admissible Mach number to |u| ≤ √3, compared with |u| ≤ √2 for LBGK, thereby improving numerical stability. A four‑velocity on‑lattice model (v₁,₂ = ±a, v₃,₄ = ±b) is derived that satisfies n* = 3, i.e., it recovers the isothermal compressible Navier‑Stokes equations exactly. By choosing a = √(3θ₀/2) and b = 3a the positivity range of r_i is maximized while keeping the lattice geometry regular.

For thermal compressible flows a five‑velocity on‑lattice set (v₁ = 0, v₂,₃ = ±a, v₄,₅ = ±b with b = 2a) is constructed. This configuration satisfies n* = 4, thus reproducing the full thermal Navier‑Stokes equations, including the correct heat flux term κ = (d + 2)νρ/2. Compared with conventional Gauss‑Hermite quadrature that requires seven velocities in one dimension (or 25–37 in two dimensions), the proposed scheme achieves the same order of accuracy with far fewer velocities, dramatically reducing computational cost.

The authors validate the theory through several benchmark problems. In a one‑dimensional shock‑tube test with a strong density ratio (ρ_L = 6, ρ_R = 1), the ζ = 4 model accurately captures the shock position (x ≈ 826) and post‑shock velocity (u ≈ 0.91) in agreement with the analytical Rankine‑Hugoniot solution. The classic LBGK model exhibits oscillations and an incorrect shock location (x ≈ 805). An entropic model remains stable but yields a slightly different velocity, reflecting its failure to satisfy the third‑order moment condition μ̂₃ = ζθ₀u.

A two‑dimensional Kelvin‑Helmholtz shear‑layer simulation further demonstrates the advantage of the parametric model. Using the tensor product of the three‑velocity ζ = 4 scheme (effectively a D2Q9‑like stencil) the authors obtain sustained vortex structures up to 2,078 time steps, whereas the standard D2Q9 LBGK becomes unstable after about 1,500 steps. Quantitative comparisons of vortex amplitude and grid‑convergence studies show that the parametric model yields lower L₂ errors across resolutions (16×16 to 64×64) than the LBGK counterpart.

The paper also discusses the practical implications of the positivity condition for r_i. By adjusting the relaxation parameter ω to match the effective viscosity ν̂ = ν(ζ − 1)/2, the discrepancy between the parametric model and the LBGK solution is reduced dramatically, confirming that the extra factor (ζ − 1)/2 compensates for the altered third‑order moment.

In summary, the “Parametric Lattice Boltzmann Method” (PLBM) provides a unified, mathematically transparent framework that subsumes existing LBGK, entropic, and Gauss‑Hermite based models. By treating discrete velocities as tunable parameters, PLBM enables designers to select the minimal number of velocities required for a desired order of accuracy while simultaneously expanding the admissible Mach and temperature ranges and enhancing numerical stability. The demonstrated reductions—from seven to five velocities for thermal compressible flows and from nine to five for isothermal two‑dimensional flows—represent a significant step toward more efficient, high‑fidelity CFD simulations of compressible and thermal phenomena. Future work may extend the approach to three‑dimensional complex geometries, multi‑phase flows, and automated parameter optimization, but the current results already establish PLBM as a powerful addition to the lattice Boltzmann toolbox.