当前用于扩散和扩散型方程的格子Boltzmann方法是否遵循最大原理和非负约束？
Do current lattice Boltzmann methods for diffusion and diffusion-type equations respect maximum principles and the non-negative constraint?
AI总结：
本文研究了当前单/多松弛时间格子Boltzmann方法在瞬态扩散型方程中是否保持最大原理、比较原理、非负约束等数学性质，并提出改进的边界条件离散方法。
摘要：
格子Boltzmann方法（LBM）已在计算流体力学中确立为有效的数值方法。最近，多松弛时间LBM被提出用于模拟各向异性对流-扩散过程。对流-扩散系统的控制微分方程已知满足最大原理、比较原理、非负约束和衰减性质。本文将证明当前单松弛时间和多松弛时间格子Boltzmann方法在瞬态扩散型方程中无法保持这些数学性质。还将证明Dirichlet边界条件的离散化会影响格子Boltzmann方法满足这些数学原理的表现。本文还提出了一种新的Dirichlet边界条件离散方法。通过几个基准问题的求解，展示了格子Boltzmann方法的性能以及边界条件离散化对上述数学性质的影响。
英文摘要：
The lattice Boltzmann method (LBM) has established itself as a valid numerical method in computational fluid dynamics. Recently, multiple-relaxation-time LBM has been proposed to simulate anisotropic advection-diffusion processes. The governing differential equations of advective-diffusive systems are known to satisfy maximum principles, comparison principles, the non-negative constraint, and the decay property. In this paper, it will be shown that current single- and multiple-relaxation-time lattice Boltzmann methods fail to preserve these mathematical properties for transient diffusion-type equations. It will also be shown that the discretization of Dirichlet boundary conditions will affect the performance of lattice Boltzmann methods in meeting these mathematical principles. A new way of discretizing the Dirichlet boundary conditions is also proposed. Several benchmark problems have been solved to illustrate the performance of lattice Boltzmann methods and the effect of discretization of boundary conditions with respect to the aforementioned mathematical properties for transient diffusion and advection-diffusion equations.
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