Solving Procedures
This implementation utilizes the Lattice Boltzmann Method (LBM) utilizing a D2Q9 lattice grid layout. Unlike macroscopic Navier-Stokes derivations, LBM resolves mesoscopic kinetic particle distributions tracking a pure, uncompromised BGK single-relaxation time step approach.
Stability Clamping: The solver prevents catastrophic interference blow-up (where $M \to c_s$) by strictly bounding calculated macroscopic variables pre-collision. This allows the BGK collision operator to naturally smooth out unbounded eddy interference without the overhead performance penalties of MRT, Smagorinsky models, or Bouzidi extrapolations.
$$ f_i(\vec{x} + \vec{c}_i \Delta t, t + \Delta t) = f_i(\vec{x}, t) - \frac{1}{\tau} \left[ f_i(\vec{x}, t) - f_i^{eq}(\vec{x}, t) \right] $$
$$ f_i^{eq} = w_i \rho \left( 1 + \frac{3 \vec{c}_i \cdot \vec{u}}{c_s^2} + \frac{9 (\vec{c}_i \cdot \vec{u})^2}{2 c_s^4} - \frac{3 \vec{u}^2}{2 c_s^2} \right) $$