Preconditioned Navier-Stokes Schemes from the Generalised Lattice Boltzmann Equation

Preconditioning of Navier-Stokes equations is a widely used technique to
speed up Computational Fluid Dynamics simulations of steady flows. In this work
a systematic study is performed of time-derivative preconditioners of Navier-Stokes
equations that can be derived from the generalized lattice Boltzmann equation. In this
way, lattice Boltzmann models equivalent to preconditioned Navier-Stokes systems are
constructed, and it becomes possible to take advantage of the knowledge generated
in this field to improve the convergence to steady state of lattice-Boltzmann flowcalculations.
The generalized lattice Boltzmann equation presents a number of tunable
parameters, which provide access to a generalized hydrodynamics. Starting from this
fully parametrized lattice Boltzmann scheme, and applying restrictions to account for
isotropy and Galilean invariance, the number of free parameters is initially reduced.
Then, with the aid of Chapman-Enskog procedure, the recovery of the Navier-Stokes
equations requires a further reduction in the number of parameters. With the final
number of free parameters, and an additional re-scaling of momentum, two different
preconditioners are obtained, which are studied according to its condition number and
compared with typical Navier-Stokes preconditioners.