Do Current Lattice Boltzmann Methods for Diffusion and Advection-Diffusion Equations Respect Maximum Principle and the Non-Negative Constraint?

Abstract

Abstract. 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 and advection-diffusion 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.

A hybrid multi-time-step framework for pore-scale and continuum-scale modeling of solute transport in porous media

“…Despite ever-growing popularity of lattice Boltzmann methods for computational fluid dynamics assumptions, these methods are prone to produce unphysical values for populations f i ; for example, see [Karimi and Nakshatrala, 2015b]. Obviously, for equation (4.1) to be meaningful, the value of population f needs to be non-negative.…”

On numerical stabilization in modeling double-diffusive viscous fingering

“…Physics dictates that quantities like concentration and (absolute) temperature can attain only non-negative values. It is now well-documented that many popular numerical methods, such as the finite element method [Liska and Shashkov, 2008;Nagarajan and Nakshatrala, 2010;Nakshatrala et al, 2013, finite volume method [Potier, 2005], and lattice Boltzmann method [Karimi and Nakshatrala, 2016], violate the maximum principle and the non-negative constraint for diffusion-type equations, especially when the diffusivity is strongly anisotropic. In this paper, we will show that numerical formulations violate the maximum principle and the non-negative constraint even under isotropic diffusivity for VF with DD effects.…”