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Stability analysis of the vectorial lattice-Boltzmann method

Kévin Guillon, Romane Hélie, Philippe Helluy

Abstract

We propose a new stability analysis of the Vectorial Lattice-Boltzmann Method (VLBM). The VLBM is a variant of the LBM with extended stability features: it allows to handle compressible flows with shock waves, while the LBM is limited to low-Mach number regime. The stability analysis is based on the Legendre transform theory. We also propose a new tool: the equivalent system analysis, which we conjecture to contains both the stability and the consistency of the VLBM.

Stability analysis of the vectorial lattice-Boltzmann method

Abstract

We propose a new stability analysis of the Vectorial Lattice-Boltzmann Method (VLBM). The VLBM is a variant of the LBM with extended stability features: it allows to handle compressible flows with shock waves, while the LBM is limited to low-Mach number regime. The stability analysis is based on the Legendre transform theory. We also propose a new tool: the equivalent system analysis, which we conjecture to contains both the stability and the consistency of the VLBM.
Paper Structure (38 sections, 8 theorems, 253 equations, 2 figures, 3 tables)

This paper contains 38 sections, 8 theorems, 253 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Hyperbolic conservation law and duality
  3. Hyperbolic systems of conservation laws
  4. Entropy and symmetrization
  5. Vectorial kinetic model
  6. Direct construction
  7. Reverse construction
  8. Splitting and over-relaxation
  9. Splitting scheme
  10. Operator notations
  11. D1Q2 case
  12. D2Q3 case
  13. D2Q4 case
  14. Transport operator in the $Y$ variables
  15. Relaxation operator in the $Y$ variables
  16. ...and 23 more sections

Key Result

Theorem 2

The change of variables $W^{*}\mapsto W(W^{*})$ symmetrizes the system of conservation laws (eq:conslaw).

Figures (2)

  • Figure 7.1: Graphic representation of the diffusive stability and hyperbolicity condition of the $D2Q3$ model for $\lambda=1$. The stable region is the blue triangle, whose vertices are the kinetic velocities.
  • Figure 7.2: Graphic representation of the diffusive stability and hyperbolicity condition of the $D2Q4$ model.

Theorems & Definitions (32)

  • Remark 1
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • proof
  • ...and 22 more