Uniform Polynomial Equations Providing Higher-order Multi-dimensional Models in Lattice Boltzmann Theory

We present a set of polynomial equations that provides models of the lattice Boltzmann theory for any required level of accuracy and for any dimensional space in a general form. We explicitly derive two- and three-dimensional models applicable to describe thermal compressible flows of the level of the Navier-Stokes equations.

💡 Research Summary

The paper introduces a unified set of polynomial equations that enables the systematic construction of lattice Boltzmann (LB) models with arbitrary accuracy and dimensionality. Starting from the continuous Maxwell‑Boltzmann distribution, the authors expand it using Taylor or Hermite polynomials, yielding a discrete equilibrium distribution of the form f_i = w_i P(v_i), where P(v) is a polynomial in the discrete velocity v_i and w_i are constant weight coefficients.

A key contribution is the formulation of moment constraints that guarantee the conservation of physical quantities (mass, momentum, pressure tensor, energy flux, etc.). These constraints are expressed as Σ_i f_i v_i^n = M_n for 0 ≤ n ≤ m, where m denotes the highest order moment that must be reproduced. By substituting the polynomial expansion, the authors derive a compact condition Σ_i w_i v_i^n = G · Γ((n+1)/2) for even n, where G involves the Gaussian Gamma function and can be written using double factorials. This condition is summarized in a single general equation (Equation 5), which must be satisfied for all even n up to 2 m.

Equation 5 provides a direct link between the desired order of accuracy (m), the dimensionality (D), and the number of independent equations that must be fulfilled. The authors introduce a combinatorial function q_D(p) that counts the number of ways to represent an integer as a sum of at most D natural numbers, allowing them to compute that 9 equations are required for a two‑dimensional Navier‑Stokes‑level model (m = 4) and 11 equations for a three‑dimensional counterpart.

Using symmetry operations (reflections about the x‑ and y‑axes and the line y = x), the authors generate full velocity sets from a small number of representative vectors. In two dimensions they obtain a 33‑velocity model; in three dimensions a 95‑velocity model is constructed. Both models have fewer discrete velocities than previously published alternatives (e.g., the 2‑D 37‑velocity and 3‑D 107‑velocity models) while achieving the same fourth‑order moment accuracy. Detailed tables list the representative velocities and associated weight coefficients.

To validate the models, the authors simulate a classic shock‑tube (Riemann) problem. Initial conditions consist of two homogeneous states with different densities and pressures separated at x = 0, with periodic boundaries in the transverse directions and open boundaries at the ends. The density and temperature profiles obtained from the 95‑velocity model match the analytical solution extremely well, especially on the plateau regions. Compared with a third‑order (m = 3) model, the fourth‑order model reproduces the shock front steepness more accurately because it captures the correct viscous effects (the analytical solution corresponds to zero viscosity, while the simulation uses a finite viscosity set by w = 1.5 × 10⁻⁴).

The authors emphasize that the framework is not limited to Navier‑Stokes accuracy. By increasing m beyond 4, the same polynomial equations can generate LB models that recover Burnett or super‑Burnett equations, opening the possibility of simulating non‑Newtonian, high‑Mach, or highly rarefied flows within the LB paradigm.

In summary, the paper provides a clear, mathematically rigorous pathway: define the desired moment order m, use Equation 5 to derive the necessary weight‑velocity constraints, and construct symmetric velocity sets that satisfy those constraints. This yields a universal, dimension‑independent method for building high‑order, low‑velocity‑count LB models, which promises significant computational savings and enhanced physical fidelity for a wide range of complex fluid‑dynamics applications.