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

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.

Figures (14)

• Figure 1: D1Q3 lattice Boltzmann scheme
• Figure 2: Evolution for an advective velocity (\ref{['u-cosinus']}) with $\, U = 0.005 \,$ and $\, N = 128 \,$ mesh points. We observe that with the cosine advection velocity, the numerical solution has no symmetry. As time progresses, we observe that the characteristic method becomes less and less precise.
• Figure 3: Unsteady evolution, constant advection field [$U = 0.05$], 64 mesh points, sinusoidal initial condition.
• Figure 4: Unsteady evolution, sinusoidal advection field [$U = 0.05$], 64 mesh points, sinusoidal initial condition.
• Figure 5: Errors measured with the maximum norm between the D1Q3 lattice Boltzmann scheme and various equivalent partial differential equations. Unsteady experiment with constant velocity field $\, U = 0.05$, finite-time evolution with final time $\, T = 1$, and initialization with a sinusoidal wave. The $x$-axis represents the number of mesh points with a logarithmic scale and the $y$-axis is graduated with the base-2 logarithm of the error. The microscopic moments are initialized with the equilibrium values.