This study analyses 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. 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 or temperature. 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 is dedicated to three aspects regarding proper inclusion of the source term in LBM schemes. 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 analyses in full detail, the implicit relation between the value of the transported 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 of common models. The second-order convergence of the proposed LBM schemes is verified on both linear and non-linear source terms. The pitfalls of the commonly used acoustic and diffusive scaling are identified and discussed. Moreover, for a simplified case, the competing errors are shown visually with isolines of error in the space of spatial and temporal resolutions.

翻译：本研究分析了一种一致恢复格点玻尔兹曼方法（LBM）二阶收敛性的方法，该方法常因源项离散不当而退化。工作聚焦于源项依赖于输运场强度的对流-扩散模型，主要结论适用于LBM框架内的多种公式体系。所有考虑的源项均被解释为分布函数零阶矩的贡献，这些贡献对应于标量场（如密度、浓度或温度）中的源项。此外，某些浸入边界方法在其公式中可被解释为源项，凸显了本工作的进一步应用前景。本文致力于LBM方案中源项正确包含的三个层面：其一，识别文献中LBM方案在源项纳入方式上观测到的差异，并通过明确代数操作阐明差异根源；其二，详尽分析输运宏观场数值与LBM密度总和之间的隐式关系，同时给出碰撞算子的三种等价形式；最后，针对多种常见模型给出该隐式关系的闭合解。所提LBM方案的二阶收敛性在线性与非线性源项上均得到验证，识别并讨论了常用声学标度与扩散标度的缺陷。此外，在简化情形下，通过空间与时间分辨率空间中的误差等值线直观展示了竞争性误差。