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:本研究分析了一种方法,以一致恢复拉蒂斯·博尔茨曼法(LBM)的二级趋同,该方法经常因所需源术语的不适当离散而退化。工作侧重于源术语取决于运输场强度的反向扩散模型。主要结果适用于LBM框架内的多种配方。所有考虑的来源术语都被解释为有助于分配功能的零移动。这些是标度字段中的来源,如密度、集中度或温度。此外,某些沉浸的边界方法可以被解释为其拟订过程中的源术语,强调这项工作的进一步应用。本文件专门论述源术语取决于运输场强度的反向扩散模型。主要结果适用于LBMF框架内的各种配方。所有考虑的来源术语都被解释为有助于分配功能的零移动。本文件中明确介绍了用于澄清所观察到的差异,并确定了其来源。第二,它详细分析了移动的直位边界方法的源值,在编程中,移动的直径直线模型的直径直线系值与直径直线模型之间的隐含关系。LBMLF 显示的平面模型显示的直径直径直方向和直径直径直径直径直线模型显示的直线模型显示的直线模型和直径直径径直方向的直径直线模型的版本是。