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This study analyses a method to consistently recover the second-orderconvergence of the lattice Boltzmann method (LBM), which is frequently degradedby the improper discretisation of required source terms. The work focuses onadvection-diffusion models in which the source terms are dependent on theintensity of transported fields. The main findings are applicable to a widerange of formulations within the LBM framework. All considered source terms areinterpreted as contributions to the zeroth-moment of the distribution function.These account for sources in a scalar field, such as density, concentration ortemperature. In addition to this, certain immersed boundary methods can beinterpreted as a source term in their formulation, highlighting a furtherapplication for this work.  This paper is dedicated to three aspects regarding proper inclusion of thesource term in LBM schemes. Firstly, it identifies the differences observedbetween the ways source terms are included in the LBM schemes present in theliterature. The algebraic manipulations are explicitly presented in this paperto clarify the differences observed, and identify their origin. Secondly, itanalyses in full detail, the implicit relation between the value of thetransported macroscopic field, and the sum of the LBM densities. Moreover,three equivalent forms of the collision operator are presented. Finally,closed-form solutions of this implicit relation are shown for a variety ofcommon models.  The second-order convergence of the proposed LBM schemes is verified on bothlinear and non-linear source terms. The pitfalls of the commonly used acousticand diffusive scaling are identified and discussed. Moreover, for a simplifiedcase, the competing errors are shown visually with isolines of error in thespace of spatial and temporal resolutions.

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