Consistency and stability of boundary conditions for a two-velocities lattice Boltzmann scheme

TL;DR

The paper analyzes boundary conditions for a simple 1D two-velocity lattice Boltzmann scheme by mapping the LBM to a corresponding Finite Difference formulation. It develops kinetic inflow Dirichlet and outflow extrapolation boundaries, complemented by boundary sources to recover second-order accuracy under certain relaxations, and it uses modified equations to study consistency. Stability is examined via GKS theory and, on coarse grids, spectral and pseudo-spectral analyses that reveal how boundary stencils, pole orders of reflection coefficients, and eigenvalue clustering influence growth and potential instabilities. The work uncovers a nuanced relationship between boundary treatment and overall scheme stability, shows that GKS-stable boundaries may degrade accuracy at initialization, and connects pole order to spectral behavior, providing insights for extending the approach to more complex velocity sets and a direct GKS framework on LBM schemes.

Abstract

We theoretically explore boundary conditions for lattice Boltzmann methods, focusing on a toy two-velocities scheme to tackle a linear one-dimensional advection equation. By mapping lattice Boltzmann schemes to Finite Difference schemes, we facilitate rigorous consistency and stability analyses. We develop kinetic boundary conditions for inflows and outflows, highlighting the trade-off between accuracy and stability, which we successfully overcome. Consistency analysis relies on modified equations, whereas stability is assessed using GKS (Gustafsson, Kreiss, and Sundstr{ö}m) theory and -- when this approach fails on coarse meshes -- spectral and pseudo-spectral analyses of the scheme's matrix that explain effects germane to low resolutions.

Figures (11)

• Figure 1: Computational mesh (dots) on $(0, T]\times (0, L)$ (in red).
• Figure 2: Modulus of the solutions of \ref{['eq:tmp7']} and corresponding $\kappa$ obtained by \ref{['eq:tmp5']}.
• Figure 3: Snapshots of the solution (first three columns) and absolute value of the solution at the left boundary node as function of time (last column). Here, we employ $C = -1/2< 0$ and $J = 1000$. Colors: $\bullet$ for $\omega = 2$, $\bullet$ for $\omega = 1.98$, $\bullet$ for $\omega = 1.96$, and $\bullet$ for $\omega = 1.94$. In the first three columns, data for $\omega<2$ are all superimposed.
• Figure 4: Snapshots of the solution (first three columns) and absolute value of the solution at the left boundary node as function of time (last column). Here, we employ $C = -1/2< 0$ and $J = 30$. Colors: $\bullet$ for $\omega = 1.98$, $\bullet$ for $\omega = 1.96$, and $\bullet$ for $\omega = 1.94$.
• Figure 5: Snapshots of the solution (first three columns) and absolute value of the solution at the left boundary node as function of time (last column). Here, we employ $C = 1/2> 0$ and $J = 30$. Color: $\bullet$ for $\omega = 1.6$

Theorems & Definitions (25)

• Theorem
• Theorem
• Proposition 1
• Remark 1: Assumptions of \ref{['prop:bulkAndInflow']}
• Proposition 2
• Conjecture 1
• Remark 2: On a possible proof of \ref{['conj:kinrod']}
• Proposition 3: Modified equation for outflow \ref{['eq:extrapolationBoundaryCondition']}
• Proposition 4: Modified equation for outflow \ref{['eq:conditionKinRod']}
• Proposition 5: $L^2$ stability conditions for \ref{['eq:bulkSchemeAbstract']} with periodicity