Monotonicity and convergence of two-relaxation-times lattice Boltzmann schemes for a non-linear conservation law

TL;DR

The paper develops a monotonicity-based convergence analysis for two-relaxation-times lattice Boltzmann schemes applied to scalar nonlinear conservation laws. By introducing two relaxation parameters and explicit equilibrium decompositions, the authors derive conditions ensuring the relaxation operator is monotone, enabling a Crandall-type proof that the numerical solution converges to the weak entropy solution. They show that TRT can reduce numerical diffusion compared to BGK while preserving stability, and provide geometric convergence to equilibrium with $L^{1}$-contraction and total-variation bounds. Numerical experiments on $ extnormal{D}_{1} extnormal{Q}_{3}$ and $ extnormal{D}_{2} extnormal{Q}_{5}$ schemes validate the theory, demonstrating improved sharpness of fronts and invariant compact sets under TRT. The results offer a rigorous, scalable framework for TRT LB schemes with practical implications for accurate, stable simulations of nonlinear conservation laws, and outline future directions toward BGK optimization, nonlinear equilibria, and systems.

Abstract

We address the convergence analysis of lattice Boltzmann methods for scalar non-linear conservation laws, focusing on two-relaxation-times (TRT) schemes. Unlike Finite Difference/Finite Volume methods, lattice Boltzmann schemes offer exceptional computational efficiency and parallelization capabilities. However, their monotonicity and $L^{\infty}$-stability remain underexplored. Extending existing results on simpler BGK schemes, we derive conditions ensuring that TRT schemes are monotone and stable by leveraging their unique relaxation structure. Our analysis culminates in proving convergence of the numerical solution to the weak entropy solution of the conservation law. Compared to BGK schemes, TRT schemes achieve reduced numerical diffusion while retaining provable convergence. Numerical experiments validate and illustrate the theoretical findings.

Figures (7)

• Figure 1: \newlabelfig:deductions0 Left: points known in $\mathcal{M}$. Right: deductions on points in $\mathcal{M}$ (colored) that we can obtain from the knowledge presented on the left, using the previous results.
• Figure 1: \newlabelfig:D1Q30 Monotonicity area $\mathcal{M}$ (in black) in the $\omega_{\mathsf{s}}\omega_{\mathsf{a}}$-plane for the $\textnormal{D}_{1}\textnormal{Q}_{3}$ scheme.
• Figure 2: \newlabelfig:D1Q3plot0 Solution of the $\textnormal{D}_{1}\textnormal{Q}_{3}$ schemes and exact solution at final time $1/4$.
• Figure 3: \newlabelfig:D1Q3_maximum0 Time-space maximum of the conserved moment for the $\textnormal{D}_{1}\textnormal{Q}_{3}$ scheme with $u^{\circ}(x) = \mathds{1}_{[0, 1/2]}(|x|)$.
• Figure 4: \newlabelfig:D1Q3convergenceEquilibrium0 Distance from the equilibrium for the $\textnormal{D}_{1}\textnormal{Q}_{3}$ scheme at different resolutions, parameters, and initial data.

Theorems & Definitions (36)

• Remark 2.1: BGK
• Remark 2.2: Magic combination
• Proposition 2.3: Consistency and modified equation
• Remark 2.4: Magic combination
• Remark 3.1: On the case $\mathscr{L}_1 = 0$: a sort of $\textnormal{D}_{d}\textnormal{Q}_{2W}$ scheme
• Definition 3.2: Monotone relaxation
• Proposition 3.3: Monotonicity conditions
• Remark 3.4: BGK vs. TRT
• Remark 3.5: On the case $\mathscr{L}_1 = 0$
• Remark 3.6: On the smoothness of the fluxes