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. •We study numerical solutions of high-order asymptotic PDEs for a lattice Boltzmann scheme.•Equivalent PDEs up to fourth-order are presented for an inhomogeneous advection problem.•Lattice Boltzmann results and a spectral approximation of the PDEs are compared.•For unsteady flow, a sufficiently high-order initialization scheme is shown to play a crucial role.