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.

翻译：本文针对一维空间非均匀平流问题，通过格子Boltzmann格式的结果处理高阶渐近等价偏微分方程的数值解。我们首先推导了各阶次的等价偏微分方程族，并将格子Boltzmann实验结果与微分方程的谱逼近结果进行比较。对于非定常情形，我们证明了微观矩足够高阶的初始化方案对于观测与逼近阶次一致的渐近误差具有关键作用。对于稳态长时间极限，我们观察到测得的渐近误差以低于渐近分析所预测的精度阶次收敛。