Fourth-order entropy-stable lattice Boltzmann schemes for hyperbolic systems

TL;DR

The paper develops a general framework for fourth-order entropy-stable lattice Boltzmann schemes for multidimensional nonlinear hyperbolic systems by combining a local relaxation (BGK/Jin–Xin) representation with a time-symmetric transport structure. Fourth-order accuracy is achieved through palindromic composition of a time-symmetric second-order LB step, yielding a practical scheme whose transport is realized by elementary grid shifts and whose relaxation can be tuned to satisfy entropy stability. Entropy stability is enforced by adaptively selecting the local relaxation rate $\omega$ to keep the microscopic entropy nonincreasing; in the linear limit, $\omega=2$ conserves entropy, and numerical tests on Burgers, shallow-water, and Euler problems demonstrate both stability and high-order convergence. The authors provide extensive numerical evidence across 1D and 2D problems, analyze L^2 stability in a linear setting, and explore variations such as equilibrium projections, highlighting practical guidelines for achieving high-order accuracy while controlling nonlinearity-driven instabilities. The work suggests promising directions for limiters, positivity preservation, and potential extensions to even higher-order schemes.”

Abstract

We present a novel framework for the development of fourth-order lattice Boltzmann schemes to tackle multidimensional nonlinear systems of conservation laws. As for other numerical schemes for hyperbolic problems, high-order accuracy applies only to smooth solutions. Our numerical schemes preserve two fundamental characteristics inherent in classical lattice Boltzmann methods: a local relaxation phase and a transport phase composed of elementary shifts on a Cartesian grid. Achieving fourth-order accuracy is accomplished through the composition of second-order time-symmetric basic schemes utilizing rational weights. This enables the representation of the transport phase in terms of elementary shifts. Introducing local variations in the relaxation parameter during each stage of relaxation ensures entropy stability of the schemes. This not only enhances stability in the long-time limit but also maintains fourth-order accuracy. To validate our approach, we conduct comprehensive testing on scalar equations and systems in both one and two spatial dimensions.

Figures (6)

• Figure 1: Way of devising a fair comparison between our new fourth-order lattice Boltzmann scheme (top) and the original second-order lattice Boltzmann scheme (bottom).
• Figure 2: Idea---illustrated in the case of a two-velocities scheme---behind the procedure selecting a variable relaxation parameter $\omega = \omega(f_+, f_-)$, in order to make the pre and the post-relaxation datum lay on the same level-set of the microscopic entropy. Notice that the equilibrium is a minimizer.
• Figure 3: Norm of the solution (top) and difference between the total microscopic entropy at time zero and eventually in time (bottom), when simulating the Burgers' equation with and without entropy conservation during the relaxation. In the bottom plot, we add $10^{-10}$ to avoid taking the logarithm of zero.
• Figure 4: Norm of the solution (height, top) and difference between the total microscopic entropy at time zero and eventually in time (bottom), when simulating the shallow water equations with and without entropy conservation during the relaxation. In the bottom plot, we add $10^{-10}$ to avoid taking the logarithm of zero.
• Figure 5: Densities at final time $T = 0.25$ with Riemann problems for the Euler equations, using Configuration 4 from lax1998solution and employing schemes (A), (B), and (C).

Theorems & Definitions (10)

• Example 1: Linear transport equation
• Example 2: Burger' equation
• Example 3: Shallow water system
• Example 4: Linear transport equation
• Remark 1
• Remark 2
• Remark 3: Cost of the scheme
• Proposition 1
• Proposition 2
• proof