A General Algorithm for Multiscale Analysis of Lattice-Boltzmann Models
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he Lattice-Boltzmann method (LBM) has emerged as a highly versatile and efficient computational tool for numerically solving partial differential equations. To comprehensively determine the macroscopic behavior of a given model in LBM, a multiscale or asymptotic analysis is indispensable. Two equivalent methods widely applied for this purpose are the Champman-Enskog analysis and Taylor expansion techniques. These analyses hold particular significance as they enable not only the determination of macroscopic equations but also the identification of high-order errors. Unfortunately, conducting such analyses can be cumbersome and prone to errors, particularly when determining high-order behaviors. In this study, we introduce a general version of the Taylor expansion analysis, which utilizes a Fourier transform to invert the series of derivatives. Additionally, we systematize the technique by defining and recognizing properties of a set of differential matrix operators. This methodology is then implemented into an algorithm based on the SymPy package, facilitating the easy determination of macroscopic equations using the equilibrium moments as inputs. We present results for both the athermal second-order model and the high-order thermal model, demonstrating agreement with previous works.
