Revisiting the second-order convergence of the lattice Boltzmann method with reaction-type source terms

Abstract

This study derives a method to consistently recover the second-order convergence of the lattice Boltzmann method (LBM), which is frequently degraded by the improper discretisation of required source terms. The work focuses on advection-diffusion models in which the source terms are dependent on the intensity of transported fields. Such terms can be observed in reaction-type equations used for example in heat and mass transfer problems or multiphase flows. The main findings are applicable to a wide range of formulations within the LBM framework. All considered source terms are interpreted as contributions to the zeroth-moment of the distribution function. These account for sources in a scalar field, such as density, concentration, temperature or a phase field. In addition to this, certain immersed boundary methods can be interpreted as a source term in their formulation, highlighting a further application for this work.This paper makes three primary contributions to the current state-of-the-art. Firstly, it identifies the differences observed between the ways source terms are included in the LBM schemes present in the literature. The algebraic manipulations are explicitly presented in this paper to clarify the differences observed, and identify their origin. Secondly, it derives in full detail, the implicit relation between the value of the transported macroscopic field, and the sum of the LBM densities. This relation is shown in the paper to be a crucial ingredient for preserving the second-order convergence in the case of complex source terms. Moreover, the derived relation is valid for any source term discretization scheme, and three equivalent forms of the second-order accurate collision operator are presented. Finally, closed form solutions of this implicit relation are shown for a variety of common models, including general linear and second order terms; population growth models, such as the Logistic or Gompertz model; and Allen-Cahn Equation.The second-order convergence of the proposed LBM schemes is verified on both linear and non-linear source terms. Commonly used diffusive and acoustic scalings are discussed, and their pitfalls are identified. Moreover, for a simplified case, the competing errors are shown visually with isolines of error in the space of spatial and temporal resolutions.

Optimization of a Micromixer with Automatic Differentiation

“…Beginning with NSE, many variants have been designed in order to solve, e.g., the ADE, multi-component flows and chemically reacting flows numerically. For the NSE and ADE, the schemes with the BGK collision operator (due to Bhatnagar, Gross and Krook [22]) have been proven to be second-order consistent for diffusive scaling (∆t ∼ ∆x 2 → 0) via asymptotic expansion [16,21,23]. Due to their inherent locality, LBM offer good parallelization properties [24], which make them attractive, especially for large-scale simulations.…”