Stability of lattice Boltzmann schemes for initial boundary value problems in raw formulation

TL;DR

The article develops a GKS-style strong stability framework for one-dimensional lattice Boltzmann MRT schemes applied to scalar hyperbolic equations with boundaries, using the raw scheme without transforming to an equivalent PDE form. It combines von Neumann analysis, characteristic polynomials, z-transform methods, and Kreiss-Lopatinskii determinants to study stability of bulk and boundary coupling, highlighting how boundary-induced instabilities can arise from shared eigenvalues or poles in boundary-eigenvectors. The work analyzes three representative schemes (D1Q2, D1Q3, and a fourth-order D1Q3 LW-type), detailing stability regions under various boundary conditions and demonstrating, via numerical simulations, how certain kinetic Dirichlet and anti-bounce-back schemes can be strongly stable, while others are not, depending on the flow orientation and stencil. These findings illuminate how boundary treatments interact with the intrinsic characteristic nature of LBMs and suggest guidelines for boundary choice in hyperbolic problems, with implications for extending GKS-style stability analyses to more complex, higher-dimensional, or nonlinear settings.

Abstract

We study the stability of one-dimensional linear lattice Boltzmann schemes for scalar hyperbolic equations with respect to boundary data. Our approach is based on the original raw algorithm on several unknowns, thereby avoiding the need for a transformation into an equivalent scalar formulation-a challenging process in presence of boundaries. To address different behaviors exhibited by the numerical scheme, we introduce appropriate notions of strong stability. They account for the potential absence of a continuous extension of the stable vector bundle associated with the bulk scheme on the unit circle for certain components. Rather than developing a general theory, complicated by the fact that discrete boundaries in lattice Boltzmann schemes are inherently characteristic, we focus on strong stability-instability for methods whose characteristic equations have stencils of breadth one to the left. In this context, we study three representative schemes. These are endowed with various boundary conditions drawn from the literature, and our theoretical results are supported by numerical simulations.

Figures (11)

• Figure 1: Sketch explaining the way schemes work, with $q = 2$ discrete velocities, $c_1 = 1$ (with associated distribution function indicated in blue), and $c_2 = -1$ (with distribution function indicated in red).
• Figure 2: Domain of dependence (blue dots) in the characteristic equation for a (science fiction) scheme with $q = 7$ discrete velocities equal to $0, \pm 1, \pm 1$, and $\pm 2$.
• Figure 3: Solution at final time for the $\text{D}_{1}\text{Q}_{2}$ scheme under $s_2 = \tfrac{3}{2}$ and $\mathscr{C} = -\tfrac{1}{2}$.
• Figure 4: Solution at final time for the $\text{D}_{1}\text{Q}_{2}$ scheme under $s_2 = 2$ and $\mathscr{C} = -\tfrac{1}{2}$.
• Figure 5: Solution at final time for the $\text{D}_{1}\text{Q}_{2}$ scheme under $s_2 = \tfrac{3}{2}$ and $\mathscr{C} = \tfrac{1}{2}$.

Theorems & Definitions (47)

• Remark 1: On $s_2, \dots, s_{q} = 0$
• Definition 1: Simple von Neumann polynomial
• Definition 2: von Neumann stability
• Definition 3: Stability of the lattice Boltzmann scheme
• Proposition 1: Stability of the lattice Boltzmann scheme
• Proposition 2: Necessary conditions for the stability of the lattice Boltzmann scheme
• Remark 2: On a reason why $s_{i} \in {[0, 2]}$ for $i\in\llbracket 2, q \rrbracket$
• Lemma 1
• proof : Proof of \ref{['prop:necConditions']}
• Definition 4: Stability of the corresponding Finite Difference scheme generated by the characteristic polynomial