Rectangular Lattice-Boltzmann Schemes with BGK-Collision Operator

The usual lattice-Boltzmann schemes for fluid flow simulations operate with square and cubic lattices. Instead of relying on square lattices it is possible to use rectangular and orthorombic lattices as well. Schemes using rectangular lattices can be constructed in several ways. Here we construct a rectangular scheme, with the BGK collision operator, by introducing 2 additional discrete velocities into the standard D2Q9 stencil and show how the same procedure can be applied in three dimensions by extending the D3Q19 stencil. The weights and scaling factors for the new stencils are found as the solutions of the well-known Hermite quadrature problem, assuring isotropy of the lattice tensors up to rank four (Philippi et al., Phys. Rev. E 73(5):056702, 2006) This isotropy is a necessary and sufficient condition for assuring the same second order accuracy of lattice-Boltzmann equation with respect to the Navier-Stokes hydrodynamic equations that is found with the standard D2Q9 and D3Q19 stencils. The numerical validation is done, in the two-dimensional case, by using the new rectangular scheme with D2R11 stencil for simulating the Taylor-Green vortex decay. The D3R23 stencil is numerically validated with three-dimensional simulations of cylindrical sound waves propagating from a point source.