Equilibrium boundary conditions for vectorial multi-dimensional lattice Boltzmann schemes

TL;DR

This work addresses the challenge of implementing numerical boundary conditions for vectorial lattice Boltzmann schemes solving hyperbolic systems of conservation laws by using equilibria driven solely by macroscopic data. It develops a $D_2Q_5$ lattice Boltzmann framework with a two-relaxation-times relaxation that enforces consistency constraints linking distributions to fluxes, and introduces equilibrium-based boundary data in ghost cells to handle space-boundaries. The authors prove convergence in the scalar case to the weak entropy solution under generalized sub-characteristic-type conditions, supported by comprehensive BV/L1 estimates, equicontinuity, and a Krushkov entropy argument, while also demonstrating that the discrete solution remains near equilibrium and converges to an equilibrium-driven limit. Numerical experiments in 1D and 2D, including Burgers and Euler equations, validate the boundary conditions’ ability to capture key physical phenomena, reveal boundary-layer behavior under wrong traces, and show applicability to both linear and nonlinear problems with boundary layers and shocks. The practical impact lies in providing a stable, convergent, and physically faithful boundary treatment for multi-dimensional LBM schemes addressing hyperbolic systems, with potential for improved accuracy and robustness in simulations of compressible flows and multi-conservation dynamics.

Abstract

The concept of equilibrium is a general tool to fill the gap between macroscopic and mesoscopic information, both within kinetic systems and kinetic schemes. This work explores the use of equilibria to devise numerical boundary conditions for multi-dimensional vectorial lattice Boltzmann schemes tackling systems of hyperbolic conservation laws. In the scalar case, we prove convergence for schemes with monotone relaxation to the weak entropy solution by Bardos, Leroux, and N{é}delec [Commun. Partial Differ. Equ., 4 (9), 1979], following the path by Crandall and Majda [Math. Comput., 34, 149 (1980)]. Numerical experiments are conducted both for scalar and vectorial problems, and demonstrate the effectiveness of equilibrium boundary conditions in capturing significant physical phenomena.

Figures (8)

• Figure 1: Solution at final time for the $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme for the transport equation with different outflow ($x=0$) boundary conditions.
• Figure 2: Solution at the iteration $n$ for the $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme with boundary conditions \ref{['eq:wrongTraceCondition']} with $\tilde{\mathsf{u}}_{\scalerel*{\circletfillhl}{\Yright}} = 1$ and zero initial data. Crosses correspond to the estimate \ref{['eq:estimationLongTimeRelaxationD1Q2WrongBC']}.
• Figure 3: Solution at final time for the $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme for the Burgers equation with different outflow ($x=0$) boundary conditions.
• Figure 4: Solution at final time for the $\textnormal{D}_{1}\textnormal{Q}_{2}$ scheme for the non-convex problem on part of the domain.
• Figure 5: Solution at final time $\mathsf{u}_{\bm{j}}^{N}$ for the $\textnormal{D}_{2}\textnormal{Q}_{4}$ for the 2D Burgers equation, for several values of $\omega$.

Theorems & Definitions (36)

• Remark 1: BGK relaxation
• Remark 2: Simpler velocity stencils
• Definition 1: Weak entropy solution
• Theorem 1: Generalized Meyers–Serrin's
• Proposition 1: Total variation of a piecewise-constant approximation
• Proposition 2: Total variation of the absolute value
• Definition 2: 2D total variation along one axis
• Remark 3
• Theorem 2
• Proposition 3