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A positive- and bound-preserving vectorial lattice Boltzmann method in two dimensions

Gauthier Wissocq, Yongle Liu, Rémi Abgrall

Abstract

We present a novel positive kinetic scheme built on the efficient collide-and-stream algorithm of the lattice Boltzmann method (LBM) to address hyperbolic conservation laws. We focus on the compressible Euler equations with strong discontinuities. Starting from the work of Jin and Xin [20] and then [4,8], we show how the LBM discretization procedure can yield both first- and second-order schemes, referred to as vectorial LBM. Noticing that the first-order scheme is convex preserving under a specific CFL constraint, we develop a blending strategy that preserves both the conservation and simplicity of the algorithm. This approach employs convex limiters, carefully designed to ensure either positivity (of the density and the internal energy) preservation (PP) or well-defined local maximum principles (LMP), while minimizing numerical dissipation. On challenging test cases involving strong discontinuities and near-vacuum regions, we demonstrate the scheme accuracy, robustness, and ability to capture sharp discontinuities with minimal numerical oscillations.

A positive- and bound-preserving vectorial lattice Boltzmann method in two dimensions

Abstract

We present a novel positive kinetic scheme built on the efficient collide-and-stream algorithm of the lattice Boltzmann method (LBM) to address hyperbolic conservation laws. We focus on the compressible Euler equations with strong discontinuities. Starting from the work of Jin and Xin [20] and then [4,8], we show how the LBM discretization procedure can yield both first- and second-order schemes, referred to as vectorial LBM. Noticing that the first-order scheme is convex preserving under a specific CFL constraint, we develop a blending strategy that preserves both the conservation and simplicity of the algorithm. This approach employs convex limiters, carefully designed to ensure either positivity (of the density and the internal energy) preservation (PP) or well-defined local maximum principles (LMP), while minimizing numerical dissipation. On challenging test cases involving strong discontinuities and near-vacuum regions, we demonstrate the scheme accuracy, robustness, and ability to capture sharp discontinuities with minimal numerical oscillations.

Paper Structure

This paper contains 25 sections, 6 theorems, 80 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Vectorial lattice Boltzmann schemes
  3. First-order scheme
  4. Second-order scheme
  5. Positive- and bound-preserving blended scheme
  6. Problem statement
  7. Construction of the blended scheme
  8. Flux limiter for density bounds
  9. Local maximum principle (LMP) based on intermediate states
  10. Relaxed local maximum principle (RLMP)
  11. Positivity preservation (PP)
  12. Relationship between the density bounds
  13. Pressure positivity
  14. Numerical results
  15. Sinusoidal entropy wave
  16. ...and 10 more sections

Key Result

Proposition 1

The Maxwellian functions of Example ex:D2Q5 are monotone nondecreasing (Bouchut criterion) if

Figures (10)

  • Figure 1: Sod shock tube at $t=0.2$ with $N=100$ points. Left: PP limiter, middle: LMP limiter, right: RLMP limiter.
  • Figure 2: Density and blending parameter ($\theta$) profiles obtained for the Shu-Osher problem Shu1988 at $t=1.8$ with $N=800$ points. Left: PP limiter, middle: LMP limiter, right: RLMP limiter. Reference: first-order scheme with $N=100000$ points.
  • Figure 3: Velocity and pressure profiles of the Shu-Osher problem obtained with the relaxed local maximum principle (RLMP) at $t=1.8$ with $N=800$ points.
  • Figure 4: Density profiles of the near vacuum smooth isentropic wave at time $t=0.1$ with $N=50$ points.
  • Figure 5: Density and blending parameter profiles of the LeBlanc problem at $t=6$ with $N=1000$ points. Left: PP limiter, middle: LMP limiter, right: RLMP limiter.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Definition 1: Bouchut criterion, from Bouchut
  • Example 1: Five-wave (D2Q5) model
  • Proposition 1
  • Remark 1
  • Example 2: First-order scheme with the D2Q5 model
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 8 more