Diffusion Lattice Boltzmann Scheme on a Orthorhombic Lattice

We present a diffusion lattice Boltzmann (DLB) scheme which is derived from first principles. As opposed to the traditional lattice BGK schemes the DLB is valid for orthorhombic lattices and it has two eigenvalues of the collision operator. It is shown that the diffusion coefficient depends only on one eigenvalue of the collision operator. Hence, the DLB scheme can be optimized with means of the additional eigenvalue of the collision operator and with different lattice spacing along the principal axes. The properties of the DLB scheme concerning consistency, stability, and accuracy are studied with eigenmode analysis. This analysis shows that the DLB scheme is consistent with diffusion for a wide range of diffusion coefficients, it has unconditional stability, and that it has third-order accuracy. Furthermore, it is shown that accuracy is improved by setting the additional eigenvalue to zero and by densifying the lattice spacing along the direction of the density gradient.