  The lattice Boltzmann method (LBM) has established itself as a valid
numerical method in computational fluid dynamics. Recently,
multiple-relaxation-time LBM has been proposed to simulate anisotropic
advection-diffusion processes. The governing differential equations of
advective-diffusive systems are known to satisfy maximum principles, comparison
principles, the non-negative constraint, and the decay property. In this paper,
it will be shown that current single- and multiple-relaxation-time lattice
Boltzmann methods fail to preserve these mathematical properties for transient
diffusion-type equations. It will also be shown that the discretization of
Dirichlet boundary conditions will affect the performance of lattice Boltzmann
methods in meeting these mathematical principles. A new way of discretizing the
Dirichlet boundary conditions is also proposed. Several benchmark problems have
been solved to illustrate the performance of lattice Boltzmann methods and the
effect of discretization of boundary conditions with respect to the
aforementioned mathematical properties for transient diffusion and
advection-diffusion equations.
