Initialisation from lattice Boltzmann to multi-step Finite Difference methods: modified equations and discrete observability

TL;DR

This paper develops a unified framework to analyze and design initialization strategies for lattice Boltzmann schemes by combining modified-equation analysis with observability concepts. By recasting LB methods as multi-step Finite Difference schemes, it derives explicit conditions under which initialization remains consistent with the target hyperbolic Cauchy problem and yields time-smooth solutions, mitigating parasitic-dissipation effects. The work provides local and prepared initialization criteria, demonstrates through 1D D1Q2 and D1Q3 examples how to achieve or fail to achieve high-order convergence, and introduces the notion of observability to identify a reduced set of initialization schemes for broad classes of LB methods, including two-relaxation-time “magic parameter” schemes. Overall, the results offer practical guidance for stable, accurate LB simulations with minimal initialization overhead and improved understanding of how initial data propagate into the bulk dynamics. These insights have potential impact on LB-based simulations in fluids and kinetic problems where initialization critically affects early-time accuracy and long-time behavior.

Abstract

Latitude on the choice of initialisation is a shared feature between one-step extended state-space and multi-step methods. The paper focuses on lattice Boltzmann schemes, which can be interpreted as examples of both previous categories of numerical schemes. We propose a modified equation analysis of the initialisation schemes for lattice Boltzmann methods, determined by the choice of initial data. These modified equations provide guidelines to devise and analyze the initialisation in terms of order of consistency with respect to the target Cauchy problem and time smoothness of the numerical solution. In detail, the larger the number of matched terms between modified equations for initialisation and bulk methods, the smoother the obtained numerical solution. This is particularly manifest for numerical dissipation. Starting from the constraints to achieve time smoothness, which can quickly become prohibitive for they have to take the parasitic modes into consideration, we explain how the distinct lack of observability for certain lattice Boltzmann schemes -- seen as dynamical systems on a commutative ring -- can yield rather simple conditions and be easily studied as far as their initialisation is concerned. This comes from the reduced number of initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods.

Figures (12)

• Figure 1: Illustration of the way of working of the lattice Boltzmann scheme (bottom) and the bulk Finite Difference scheme (top). The former acts both on the conserved (light violet) and the non-conserved (dark violet) moments. The latter implies only the conserved moment, drawn in light violet in the initialisation layer and in red in the bulk. Remark that to compute the conserved moment for the bulk Finite Difference scheme at time $(Q + 2)\Delta t$, one can either rely on the information at time $(Q + 1)\Delta t$ in light violet (from the lattice Boltzmann scheme) or on the one in red (from the Finite Difference scheme), as highlighted by the symbol $\blacksquare$. This holds because these quantities are equal for any time step in the bulk, for they stem from a common initialisation process. Partial transparency is used to denote the initialisation steps.
• Figure 2: Example of behaviour of the inner expansion (dots, concerning the starting schemes) and the outer expansion (dashed lines, relative to the bulk Finite Difference scheme) at different orders in $\Delta x$ for $\Delta x \to 0$.
• Figure 3: $L^2$ errors at the final time for two forward centered initialisations \ref{['eq:CenteredGood']} (top) and \ref{['eq:CenteredBad']} (bottom). Since the letter irremediably perturbs the conserved moment feeding the bulk Finite Difference scheme, the orders of convergence above one are lowered. Color legend for the relaxation parameter $s_2$: $\bullet$ for $s_2 = 1.1$, $\bullet$ for $s_2 = 1.2$, $\bullet$ for $s_2 = 1.4$, $\bullet$ for $s_2 = 1.6$, $\bullet$ for $s_2 = 1.8$, and $\bullet$ for $s_2 = 2$.
• Figure 4: Test for the smoothness in time close to $t = 0$ for $s_2 = 1.99$: difference between exact and numerical solution at the eighth lattice point.
• Figure 5: Test for the smoothness in time close to $t = 0$ for $s_2 = 2$: difference between exact and numerical solution at the eighth lattice point. Compared to \ref{['fig:InitialError']}, CR0 and CR1 from van2009smooth cannot be used.

Theorems & Definitions (35)

• Example 1: $\textrm{D}_{1}\textrm{Q}_{2}$
• Remark 1
• Definition 1: Asymptotic equivalence
• Example 2
• Theorem 1: bellotti2021equivalentequations Modified equation of the bulk scheme
• Proposition 1: Modified equation of the starting schemes with local initialisation
• proof
• Corollary 1: Consistency of the starting schemes with local initialisation
• proof : Proof of \ref{['prop:LocalInitialisation']}
• Proposition 2: Modified equation of the starting schemes with prepared initialisation