@Article{CiCP-20-2,
author = {},
title = {Do Current Lattice Boltzmann Methods for Diffusion and Advection-Diffusion Equations Respect Maximum Principle and the Non-Negative Constraint?},
journal = {Communications in Computational Physics},
year = {2016},
volume = {20},
number = {2},
pages = {374--404},
abstract = {<p style="text-align: justify;">The Lattice Boltzmann Method (LBM) has established itself as a popular
numerical method in computational fluid dynamics. Several advancements have been
recently made in LBM, which include multiple-relaxation-time LBM to simulate anisotropic
advection-diffusion processes. Because of the importance of LBM simulations
for transport problems in subsurface and reactive flows, one needs to study the accuracy
and structure preserving properties of numerical solutions under the LBM. The
solutions to advective-diffusive systems are known to satisfy maximum principles,
comparison principles, the non-negative constraint, and the decay property. In this
paper, using several numerical experiments, 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. We will also show that these
violations may not be removed by simply refining the discretization parameters. More
importantly, it will be shown that meeting stability conditions alone does not guarantee
the preservation of the aforementioned mathematical principles and physical
constraints in the discrete setting. A discussion on the source of these violations and
possible approaches to avoid them is included. A condition to guarantee the non-negativity
of 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 research
problems, which should guide researchers to develop LBM-based formulations
for transport problems that respect important mathematical properties and physical
constraints in the discrete setting. This paper can also serve as a good source of benchmark
problems for such future research endeavors. Second, this study cautions the
practitioners of the LBM for transport problems with the associated numerical deficiencies of the LBM, and provides guidelines for performing predictive simulations of
advective-diffusive processes using the LBM.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.181015.270416a},
url = {https://global-sci.com/article/80202/do-current-lattice-boltzmann-methods-for-diffusion-and-advection-diffusion-equations-respect-maximum-principle-and-the-non-negative-constraint}
}