Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation

A lattice Boltzmann model with a multiple-relaxation-time (MRT) collision operator for the convection–diffusion equation is presented. The model uses seven discrete velocities in three dimensions (D3Q7 model). The off-diagonal components of the relaxation-time matrix, which originate from the rotation of the principal axes, enable us to take into account full anisotropy of diffusion. An asymptotic analysis of the model equation with boundary rules for the Dirichlet and Neumann-type (specified flux) conditions is carried out to show that the model is first- and second-order accurate in time and space, respectively. The results of the analysis are verified by several numerical examples. It is also shown numerically that the error of the MRT model is less sensitive to the variation of the relaxation-time coefficients than that of the classical BGK model. In addition, an alternative treatment for the Neumann-type boundary condition that improves the accuracy on a curved boundary is presented along with a numerical example of a spherical boundary.