Do Current Lattice Boltzmann Methods for Diffusion and Advection-Diffusion Equations Respect Maximum Principle and the Non-Negative Constraint? (Communications in Computational Physics, 2018)
The Lattice Boltzmann Method (LBM) has established itself as a popularnumerical method in computational fluid dynamics. Several advancements have beenrecently made in LBM, which include multiple-relaxation-time LBM to simulate anisotropicadvection-diffusion processes. Because of the importance of LBM simulationsfor transport problems in subsurface and reactive flows, one needs to study the accuracyand structure preserving properties of numerical solutions under the LBM. Thesolutions to advective-diffusive systems are known to satisfy maximum principles,comparison principles, the non-negative constraint, and the decay property. In thispaper, using several numerical experiments, it will be shown that current single- andmultiple-relaxation-time lattice Boltzmann methods fail to preserve these mathematicalproperties for transient diffusion-type equations. We will also show that theseviolations may not be removed by simply refining the discretization parameters. Moreimportantly, it will be shown that meeting stability conditions alone does not guaranteethe preservation of the aforementioned mathematical principles and physicalconstraints in the discrete setting. A discussion on the source of these violations andpossible approaches to avoid them is included. A condition to guarantee the non-negativityof concentration under LBM in the case of isotropic diffusion is also derived.The impact of this research is twofold. First, the study poses several outstanding researchproblems, which should guide researchers to develop LBM-based formulationsfor transport problems that respect important mathematical properties and physicalconstraints in the discrete setting. This paper can also serve as a good source of benchmarkproblems for such future research endeavors. Second, this study cautions thepractitioners of the LBM for transport problems with the associated numerical deficiencies of the LBM, and provides guidelines for performing predictive simulations ofadvective-diffusive processes using the LBM.