Bruce M Boghosian | Searcharxiv

arXiv2025

General fourth order Chapman-Enskog expansion of lattice Boltzmann schemes

In order to derive the equivalent partial differential equations of a lattice Boltzmann scheme,the Chapman Enskog expansion is very popular in the lattive Boltzmann community. A maindrawback of this approach is the fact that multiscale expansions are used without any clearmathematical signification of the various variables and operators. Independently of thisframework, the Taylor expansion method allows to obtain formally the equivalent partialdifferential equations. In this contribution, we prove that both approaches give identicalresults with acoustic scaling for a very general family of lattice Boltzmann schemes and upto fourth order accuracy. Examples with a single scalar conservation illustrate our purpose.

math.NA↗

arXiv2024

Numerical approximations of a lattice Boltzmann scheme with a family of partial differential equations

In this contribution, we address the numerical solutions of high-order asymptotic equivalent partial differential equations with the results of a lattice Boltzmann scheme for an inhomogeneous advection problem in one spatial dimension. We first derive a family of equivalent partial differential equations at various orders, and we compare the lattice Boltzmann experimental results with a spectral approximation of the differential equations. For an unsteady situation, we show that the initialization scheme at a sufficiently high order of the microscopic moments plays a crucial role to observe an asymptotic error consistent with the order of approximation. For a stationary long-time limit, we observe that the measured asymptotic error converges with a reduced order of precision compared to the one suggested by asymptotic analysis.

math.NA↗