Viscosity independent numerical errors for Lattice Boltzmann models: From recurrence equations to “magic” collision numbers

Abstract

We prove for generic steady solutions of the Lattice Boltzmann (LB) models that the variation of the numerical errors is set by specific combinations (called “magic numbers”) of the relaxation rates associated with the symmetric and anti-symmetric collision moments. Given the governing dimensionless physical parameters, such as the Reynolds or Peclet numbers, and the geometry of the computational mesh, the numerical errors remain the same for any change of the transport coefficients only when the “free” (“kinetic”) anti-symmetric rates and the boundary rules are chosen properly. The single-relaxation-time (BGK) model has no free collision rate and yields viscosity dependent errors with any boundary scheme for hydrodynamic problems. The simplest and most efficient collision operator for invariant errors is the two-relaxation-times (TRT) model. As an example, this model is able to compute viscosity independent permeabilities for any porous structure. These properties are derived from steady recurrence equations, obtained through linear combinations of the LB evolution equations, in which the equilibrium and non-equilibrium components are directly interconnected via finite-difference link-wise central operators. The explicit dependency of the non-equilibrium solution on the relaxation rates is then obtained. This allows us, first, to confirm the governing role of the “magic” combinations for steady solutions of the Stokes equation, second, to extend this property to steady solutions of the Navier–Stokes and anisotropic advection–diffusion equations, third, to develop a parametrization analysis of the microscopic and macroscopic closure relations prescribed via link-wise boundary schemes.