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Finite Difference formulation of any lattice Boltzmann scheme

Thomas Bellotti, Benjamin Graille, Marc Massot

TL;DR

This work shows that any lattice Boltzmann scheme can be reformulated as a multi-step Finite Difference scheme on the conserved variables by embedding LB operators in a commutative FD-ring and applying the Cayley–Hamilton theorem. This yields a rigorous FD-consistency notion for LB methods and proves that von Neumann stability for LB is equivalent to the stability of its FD counterpart in the linear-equilibrium setting, enabling standard FD tools for LB analysis. The authors develop explicit constructions for one or multiple conserved moments, discuss initialization schemes, and illustrate the framework with examples such as D1Q3, linking LB formulations to well-established FD theory and Lax–Richtmyer convergence results. Overall, the paper provides a unifying mathematical foundation that connects LB schemes with FD multi-step methods, with significant implications for stability, convergence, and the design of LB algorithms.

Abstract

Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified by means of numerical illustrations.

Finite Difference formulation of any lattice Boltzmann scheme

TL;DR

This work shows that any lattice Boltzmann scheme can be reformulated as a multi-step Finite Difference scheme on the conserved variables by embedding LB operators in a commutative FD-ring and applying the Cayley–Hamilton theorem. This yields a rigorous FD-consistency notion for LB methods and proves that von Neumann stability for LB is equivalent to the stability of its FD counterpart in the linear-equilibrium setting, enabling standard FD tools for LB analysis. The authors develop explicit constructions for one or multiple conserved moments, discuss initialization schemes, and illustrate the framework with examples such as D1Q3, linking LB formulations to well-established FD theory and Lax–Richtmyer convergence results. Overall, the paper provides a unifying mathematical foundation that connects LB schemes with FD multi-step methods, with significant implications for stability, convergence, and the design of LB algorithms.

Abstract

Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified by means of numerical illustrations.
Paper Structure (25 sections, 16 theorems, 83 equations, 4 figures, 1 algorithm)

This paper contains 25 sections, 16 theorems, 83 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The example of Ordinary Differential Equations
  3. Algebraic form of lattice Boltzmann schemes
  4. Spatial and temporal discretization
  5. Discrete velocities and shift operators
  6. Lattice Boltzmann algorithm: collide and stream
  7. Collision phase
  8. Stream phase
  9. Monolithic scheme
  10. Main result of the paper
  11. Characteristic polynomial and Cayley-Hamilton theorem
  12. Corresponding Finite Difference schemes
  13. One conserved moment
  14. Several conserved moments and vectorial schemes
  15. Initialization schemes
  16. ...and 10 more sections

Key Result

Proposition 1

$(\mathcal{T}^{d}_{\Delta x}, \circ)$ forms an Abelian group.

Figures (4)

  • Figure 1: Maximal space-time domain of dependence of the corresponding Finite Difference scheme for $N = 1$ (full black points inside the grey area) by virtue of \ref{['prop:ReductionFiniteDifferenceGeneral']} in the case of $d = 1$. The maximal space-time slopes are determined by the maximal shift of the considered scheme whereas the number of involved time-steps is at most $q + 1$.
  • Figure 2: Stability region (in black), obtained numerically, as function of $s$ and $\mathtt{D}$ for the $\text{D}_{1}\text{Q}_{3}$ of \ref{['ex:D1Q3OneConservedVariable']}, considering $\mathtt{C} = 1/2$. The black dashed line corresponds to $\mathtt{D} = 3\mathtt{C}^2 /2 - 1 = -0.625$, for which the residual diffusivity vanishes, see \ref{['eq:D1Q3NumericalTestEquivalentEquation']}. The right image is a magnification of the left one close to $s = 1.2$.
  • Figure 3: $\mathtt{D} = 0.4$. $\ell^2$ error at final time $T$ between the solution (conserved moment) of lattice Boltzmann scheme and the exact solution, for different initial data (a), (b), (c) and (d) and different relaxation parameters $s$.
  • Figure 4: $\mathtt{D} = -0.625$. $\ell^2$ error at final time $T$ between the solution (conserved moment) of lattice Boltzmann scheme and the exact solution, for different initial data (a), (b), (c) and (d) and different relaxation parameters $s$.

Theorems & Definitions (43)

  • Definition 1: Shift operator
  • Definition 2: Product
  • Proposition 1
  • Definition 3: Finite Difference operators
  • Proposition 2: Ring of Finite Difference operators
  • Remark 1
  • Remark 2
  • Example 1: $\text{D}_{1}\text{Q}_{3}$ scheme with one conserved moment
  • Definition 4: Characteristic polynomial
  • Example 2
  • ...and 33 more