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On Lattice Boltzmann Methods based on vector-kinetic models for hyperbolic partial differential equations

Megala Anandan, S. V. Raghurama Rao

TL;DR

The paper addresses robust numerical treatment of hyperbolic conservation laws using lattice Boltzmann methods derived from vector-kinetic models. It compares explicit and semi-implicit discretisations, derives a macroscopic finite difference form, and proves entropy-consistent properties, TVB, and positivity for the semi-implicit scheme, while detailing sub-characteristic constraints for the explicit variant. It extends the LBM to hyperbolic systems with source terms through well balanced r_q modelling and Crank-Nicolson discretisation, and introduces a D2Q9 diagonal upwind LBM that captures multidirectional fluxes. The work provides explicit equilibrium constructions (D1Q2, D1Q3, upwind DdQ) and boundary/corner treatments for D2Q9, supported by comprehensive 1D, 2D and 3D numerical benchmarks. Overall, the framework yields stable, accurate, well balanced LBMs for hyperbolic PDEs with or without sources and enables enhanced multidimensional upwinding through diagonal velocities.

Abstract

In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a D$2$Q$9$ model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.

On Lattice Boltzmann Methods based on vector-kinetic models for hyperbolic partial differential equations

TL;DR

The paper addresses robust numerical treatment of hyperbolic conservation laws using lattice Boltzmann methods derived from vector-kinetic models. It compares explicit and semi-implicit discretisations, derives a macroscopic finite difference form, and proves entropy-consistent properties, TVB, and positivity for the semi-implicit scheme, while detailing sub-characteristic constraints for the explicit variant. It extends the LBM to hyperbolic systems with source terms through well balanced r_q modelling and Crank-Nicolson discretisation, and introduces a D2Q9 diagonal upwind LBM that captures multidirectional fluxes. The work provides explicit equilibrium constructions (D1Q2, D1Q3, upwind DdQ) and boundary/corner treatments for D2Q9, supported by comprehensive 1D, 2D and 3D numerical benchmarks. Overall, the framework yields stable, accurate, well balanced LBMs for hyperbolic PDEs with or without sources and enables enhanced multidimensional upwinding through diagonal velocities.

Abstract

In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a DQ model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.
Paper Structure (41 sections, 5 theorems, 112 equations, 8 figures, 2 tables)

This paper contains 41 sections, 5 theorems, 112 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Hyperbolic conservation law
  4. Vector-kinetic equation
  5. Lattice Boltzmann equation
  6. Chapman-Enskog expansion
  7. Equilibrium function
  8. Classical D$1$Q$2$
  9. D$1$Q$3$
  10. Upwind D$\overline{d}$Q$(2\overline{d}+1)$
  11. Properties of the lattice Boltzmann equation
  12. H-inequality
  13. Macroscopic finite difference form
  14. Consistency
  15. Total Variation Boundedness
  16. ...and 26 more sections

Key Result

Theorem 1

There exists an inequality corresponding to the semi-implicit discretisation Ch_LB_H_fexp of VKE with $\omega=\frac{\Delta t}{\epsilon}> 0$. Here, $H_q(f_q)$ follows the definition Ch_LB_Hineq_def.

Figures (8)

  • Figure 1: Plot of $\Tilde{\omega}$ vs. $\omega$
  • Figure 2: Boundary conditions (Black lines indicate boundaries; red arrows indicate unknown functions at each boundary)
  • Figure 3: Corner conditions (Red arrows indicate unknown functions that are evaluated; Blue arrows indicate unknown functions that are not evaluated)
  • Figure 4: Top: Extended MacCormack's method with limiter based on $U^n$(Reproduced from LeVequeYee1990), Bottom: Our LB scheme for hyperbolic conservation laws with source terms
  • Figure 5: Cross-sectional plot at $x_2=0$. Top: Extended MacCormack's method with limiter based on $U^n$, Bottom: Formulated LB scheme for hyperbolic conservation laws with source terms
  • ...and 3 more figures

Theorems & Definitions (26)

  • Remark 1
  • Remark 2
  • Definition 1
  • Theorem 1
  • proof
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof
  • Remark 5
  • ...and 16 more