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

TL;DR

This paper investigates the numerical solutions of high-order asymptotic PDEs derived from a lattice Boltzmann scheme for inhomogeneous advection, emphasizing the importance of initialization and analyzing long-term errors.

Contribution

It introduces a family of equivalent PDEs at various orders and compares lattice Boltzmann results with spectral approximations, highlighting the role of initialization in unsteady cases.

Findings

Initialization at high order affects asymptotic error in unsteady simulations.

Measured long-time asymptotic error converges at a reduced order.

Spectral approximation aligns with lattice Boltzmann results for PDEs.

Abstract

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.