The scalar field and the non-equilibrium solutions of the linear advection-diffusion d2Q9 Lattice Boltzmann (LBM) two-relaxation-times (TRT) scheme are constructed analytically. The scheme copes with an infinite number of suitable, second-order accurate, equilibrium weights. Here, the simplest, translation-invariant geometry with an implicitly located, straight or diagonal, grid-aligned interface (boundary) is addressed. We show that these two interface (boundary) orientations are accommodated with the help of two distinctive, anisotropic, discrete-exponential algebraic solution components, referred to as the A-layer and the B-layer. Being unpredicted by the perturbative analysis, such as the Chapman-Enskog, asymptotic or truncation, their solution is derived symbolically from the TRT recurrence equations, subject to the local mass conservation solvability and effective closure conditions. When the interface (boundary) is “diagonal”, the A-layer perturbs the simplest physical solutions, like the piece-wise linear, polynomial or exponential scalar field, rendering the macroscopic solution weight-dependent and delaying its convergence to the first order; the A-layer base depends upon the weights, free relaxation parameter Λ and physical numbers. In contrast, the B-layer, invisible to the scalar field, typically accommodates the non-equilibrium discrepancy between the normal and diagonal directions on the “straight” interface (boundary); the B-layer base is fixed by Λ alone. The A-layer and B-layer may coexist and degrade the physical solution gradient and its convergence. Only the D2Q5 model is free from all these effects in the straight and diagonal orientations, while the diagonally-rotated D2Q5 model is unsuitable because of the “checkerboard” effect. These spurious corrections are not the Knudsen layers, but they present the LBM response for any-order bulk mismatch with the implicit or explicit interface (boundary) treatment; the A-layer and B-layer bring them in evidence and provide excellent benchmarks for their attenuation through interface-conjugate or adaptive refinement techniques. Our approach extends to any lattice, linear collision, source term, heterogeneity and LBM problem class.

分析地构造了线性对流扩散d2Q9格子Boltzmann（LBM）两次弛豫时间（TRT）格式的标量场和非平衡解。该方案可应对无数个合适的二阶精确平衡权重。在这里，解决了最简单的，平移不变的几何结构，该结构具有隐式定位的直线或对角线，网格对齐的界面（边界）。我们表明，这两个界面（边界）方向是在两个独特的，各向异性的，离散指数的代数解分量（称为A层和B层）的帮助下获得的。由于不受Chapman-Enskog，渐近或截断之类的扰动分析的影响，它们的解是从TRT递归方程式中得出的，须遵守当地群众的节约性可溶解性和有效的封闭条件。当界面（边界）为“对角线”时，A层会扰动最简单的物理解，例如分段线性，多项式或指数标量场，使宏观解的权重相关并将其收敛到一阶。A层的基底取决于重量，自由松弛参数Λ和物理数。相比之下，标量场不可见的B层通常会适应“直线”界面（边界）上法线方向和对角线方向之间的非平衡差异。B层的基底仅由Λ固定。A层和B层可以共存并降低物理溶液梯度及其收敛。只有D2Q5模型在直线和对角线方向上没有所有这些效果，而对角旋转的D2Q5模型由于“棋盘”效果而不合适。这些杂散校正不是Knudsen层，但是它们通过隐式或显式接口（边界）处理呈现了针对任何顺序的体积不匹配的LBM响应；A层和B层为它们提供了证据，并通过界面共轭或自适应细化技术为其衰减提供了出色的基准。我们的方法扩展到任何晶格，线性碰撞，源项，异质性和LBM问题类别。但是他们通过隐式或显式接口（边界）处理呈现了对于任何顺序的批量不匹配的LBM响应；A层和B层为它们提供了证据，并通过界面共轭或自适应细化技术为其衰减提供了出色的基准。我们的方法扩展到任何晶格，线性碰撞，源项，异质性和LBM问题类别。但是他们通过隐式或显式接口（边界）处理呈现了对于任何顺序的批量不匹配的LBM响应；A层和B层为它们提供了证据，并通过界面共轭或自适应细化技术为其衰减提供了出色的基准。我们的方法扩展到任何晶格，线性碰撞，源项，异质性和LBM问题类别。