Lagrangian Entropic Lattice Boltzmann Method (LELBM)

• LELBM is a compressible, thermal, and moment-based lattice Boltzmann framework that uses entropy minimization and adaptive Lagrangian stencils to simulate high-speed, supersonic flows.
• It integrates regularized central moment collision with a double-distribution function for polyatomic gases to independently control shear, bulk, and thermal transport.
• The method employs local moment streaming and entropic population reconstruction to overcome traditional velocity restrictions, ensuring explicit time stepping and memory efficiency.

Searching arXiv for the LELBM paper and closely related entropic LBM work. Lagrangian Entropic Lattice Boltzmann Method (LELBM) is a compressible, thermal, moment-based, entropy-stabilized lattice Boltzmann framework developed for high-speed and supersonic flow simulation. In the formulation introduced by Fei and Luo, LELBM combines regularized central moment collision, a double-distribution function for polyatomic gases, adaptive local multispeed shifted velocity sets called Lagrangian acoustic stencils, entropy-based population reconstruction, and moment streaming, with the stated objective of overcoming conventional restrictions on velocity, temperature, viscosity, positivity, stability, and memory while retaining explicit time stepping (Noh et al., 9 Aug 2025).

1. Definition and scope

LELBM was introduced as a “Courant-free supersonic compressible flow simulation” framework within lattice Boltzmann methodology. Its target regime is explicitly compressible, thermal, and supersonic flow, rather than the low-Mach, near-isothermal settings for which fixed-lattice lattice Boltzmann schemes are usually optimized. The method is presented as a unified construction in which transport remains explicit, collision is performed in central moment space, and post-collision populations are reconstructed by a discrete entropy minimization procedure that enforces positivity and prescribed moments simultaneously (Noh et al., 9 Aug 2025).

The qualifier “Lagrangian” refers primarily to transport along particle characteristics, the use of relative velocities $\bm{c}_i-\bm{u}$, local velocity sets centered on the bulk motion, and the allowance of large displacements per timestep. In this sense, LELBM departs from the conventional interpretation of lattice Boltzmann streaming as a low-Courant fixed-lattice shift. The paper argues that the standard explicit-CFD Courant number,

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$

does not directly constrain its streaming step, because populations or moment fractions are transported collisionlessly to landing cells, and stability is governed mainly by successful reconstruction rather than a classical CFL restriction (Noh et al., 9 Aug 2025).

Within the broader entropic lattice Boltzmann literature, LELBM is best understood as a direct Lagrangian formulation built on ideas that earlier works had developed separately. Those earlier works analyzed entropy-minimized equilibria, entropy-controlled collision, adaptive entropic relaxation, moving-boundary entropic solvers, and multilevel entropic coupling, but did not formulate a Lagrangian entropic compressible method with adaptive local stencils and moment-only storage. This suggests that LELBM is both a continuation of entropic lattice Boltzmann theory and a distinct algorithmic synthesis (Atif et al., 2022).

2. Kinetic foundation and thermodynamic structure

The starting point is the continuous Boltzmann equation

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$

with BGK collision

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$

After discretization along characteristics, the method adopts a transport-collision form in which the post-collision state is streamed along discrete velocities. The discrete population is interpreted as a velocity-space integral over a small cell around $\bm{c}_i$,

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$

and, in the LELBM presentation, also in a finite-volume sense as a cell-volume-integrated quantity (Noh et al., 9 Aug 2025).

For polyatomic compressible flow, LELBM uses a double-distribution function (DDF) formulation. The two distributions are “fluons” $f_i$ for translational dynamics and “phonons” $g_i$ for internal degrees of freedom. They arise from reducing a $(D+K)$-dimensional monatomic equilibrium to two $D$-dimensional distributions,

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$0

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$1

The corresponding ideal-gas thermodynamics are controlled through the internal degrees of freedom $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$2,

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$3

so that $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$4 sets the specific heat ratio (Noh et al., 9 Aug 2025).

The equilibrium distribution remains Gaussian in continuous velocity space,

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$5

with the full polyatomic form

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$6

Macroscopic fields are extracted from raw moments,

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$7

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$8

with pressure and sound speed

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$9

These relations make LELBM an ideal-gas, calorically perfect gas model with explicit control of thermal state and acoustic scale (Noh et al., 9 Aug 2025).

The entropic content of the method resides in reconstruction rather than in a classical ELBM path-length search. The continuous $$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$0-functional is

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$1

with entropy relation $$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$2. In discrete form, normalized post-collision populations are reconstructed by minimizing the discrete entropy approximation

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$3

subject to moment constraints. Because the minimizer has exponential form, reconstructed populations are strictly positive (Noh et al., 9 Aug 2025).

3. Central-moment collision and transport coefficients

LELBM performs collision in central moment space. This is a numerically consequential design choice because, in high-speed flow, raw moments contain large bulk-motion contributions, whereas the relevant non-equilibrium content may be comparatively small. Raw moments are defined by

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$4

while central moments use relative velocities,

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$5

Typical raw-central transforms include

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$6

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$7

Non-equilibrium central moments are formed by subtracting equilibrium central moments from these quantities (Noh et al., 9 Aug 2025).

The second-order non-equilibrium central moments are split into shear and bulk components,

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$8

$$\frac{\partial f}{\partial t} + \xi_\alpha \frac{\partial f}{\partial x_\alpha} + \frac{F_\alpha}{\rho}\frac{\partial f}{\partial \xi_\alpha} = \Omega(f),$$9

Post-collision moments are then prescribed by separate relaxation of shear, bulk, and thermal channels:

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$0

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$1

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$2

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$3

An additional imposed moment is

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$4

This structure is the basis for independent control of transport processes (Noh et al., 9 Aug 2025).

The Chapman–Enskog analysis reported for the method yields separate constitutive coefficients:

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$5

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$6

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$7

Using the discrete collision frequencies

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$8

these become

$\Omega(f) = -\frac{1}{\tau}(f-f^{\mathrm{eq}}).$9

$\bm{c}_i$0

$\bm{c}_i$1

Accordingly, shear viscosity, bulk viscosity, and thermal conductivity are controlled by $\bm{c}_i$2, $\bm{c}_i$3, and $\bm{c}_i$4, respectively. The reported continuum limit is the compressible Navier–Stokes–Fourier system, with viscous stress

$\bm{c}_i$5

This explicitly distinguishes LELBM from simpler compressible lattice Boltzmann schemes in which transport parameters are not independently tunable (Noh et al., 9 Aug 2025).

A plausible implication is that LELBM inherits some conceptual continuity with earlier entropic MRT and KBC-style constructions, which also separated hydrodynamically relevant and higher-order modes for stability purposes. However, the specific LELBM formulation is not a direct reuse of those earlier schemes; its collision is embedded in a distinct compressible DDF, adaptive-stencil, entropic-reconstruction framework (Dorschner et al., 2016).

4. Lagrangian acoustic stencils and entropic population reconstruction

One of the central novelties of LELBM is the construction of local, adaptive, shifted multispeed velocity sets called Lagrangian acoustic stencils (LAS). The motivation is explicit: a fixed lattice sized for the hottest or fastest region everywhere is memory-expensive and may still fail locally, whereas the local equilibrium is a Gaussian centered at $\bm{c}_i$6 with width $\bm{c}_i$7, so the discrete velocities should be centered near $\bm{c}_i$8 and extend over a radius proportional to $\bm{c}_i$9 (Noh et al., 9 Aug 2025).

The method defines the conformation number

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$0

the lattice Courant number

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$1

and the interlattice Courant number

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$2

The acoustic radius is

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$3

and the local stencil contains all velocities satisfying

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$4

The implementation reported in the paper uses $$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$5, corresponding to approximately 99.99% Gaussian coverage. The minimum radius $$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$6 is imposed to maintain enough points for third-order moment reconstruction, and the asymptotic number of stencil points is stated as

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$7

This construction is the source of the method’s large effective lattice Courant numbers and local adaptivity (Noh et al., 9 Aug 2025).

At low temperature, uniform velocity spacing under-resolves the narrow equilibrium peak. The method therefore introduces internal velocity-space refinement with

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$8

and finer spacing within regions satisfying

$$f_i(\bm{x},\bm{c}_i)=\iiint_{\Delta C_i} f(\bm{x},\bm{\xi})\,d\bm{\xi},$$9

with velocity-cell volumes scaling as

$f_i$0

The authors report that refinement improves moderately low-temperature behavior, but that entropic population reconstruction still tends to fail below approximately $f_i$1. This low-temperature limit is one of the clearest stated algorithmic constraints (Noh et al., 9 Aug 2025).

Entropic population reconstruction (EPR) converts prescribed post-collision moments into actual post-collision populations. Direct matrix inversion is described as undesirable because it requires a square $f_i$2 system, overconstrains the problem, and often produces negative populations. EPR instead solves a maximum-entropy problem. For $f_i$3, the stationary condition is

$f_i$4

subject to

$f_i$5

The solution is

$f_i$6

with an analogous form for $f_i$7. Positivity follows immediately:

$f_i$8

The Lagrange multipliers are obtained by Newton iteration,

$f_i$9

with

$g_i$0

These matrices are described as dense but small, symmetric, and positive definite, and the implementation uses Cholesky decomposition (Noh et al., 9 Aug 2025).

Preconditioning is supplied by a local Boltzmann approximation,

$g_i$1

with initial multipliers

$g_i$2

The paper reports convergence in fewer than five iterations in typical cases. If reconstruction succeeds, the stated guarantees are positivity, exact satisfaction of the prescribed moment constraints, and entropy-consistent post-collision populations. The paper also states that reconstruction failure is mostly due to insufficient local stencil support, which is why LAS and EPR are designed as coupled components (Noh et al., 9 Aug 2025).

This entropic reconstruction mechanism differs from earlier entropic lattice Boltzmann formulations in which entropy primarily determined an involution parameter or adaptive relaxation path. In LELBM, the entropy principle is moved directly into the reconstruction of populations from moments, which is a different algorithmic role for entropy (Hosseini et al., 2023).

5. Moment streaming, explicitness, and computational organization

LELBM stores and transports moments rather than full populations. This “moment streaming” (MS) algorithm is the method’s main memory-reduction device. Conventional multispeed thermal lattice Boltzmann methods store all populations; the LELBM paper notes that a three-dimensional $g_i$3 model for polyatomic flow would require at least 686 stored populations per cell. By contrast, LELBM stores only the moment set required by the Navier–Stokes–Fourier closure (Noh et al., 9 Aug 2025).

The stored raw moments comprise one zeroth-order moment, $g_i$4 first-order moments, $g_i$5 second-order moments, and $g_i$6 contracted third-order moments, together with DDF moments $g_i$7. In three dimensions this gives 13 fluon moments and 4 phonon moments, for a total of 17 stored variables per cell. The paper states that translational memory scales as

$g_i$8

plus

$g_i$9

for the polyatomic contribution. This is contrasted with the temperature-dependent population count

$(D+K)$0

required by explicit storage of local multispeed populations (Noh et al., 9 Aug 2025).

Streaming is performed by reconstructing temporary post-collision populations locally, converting them immediately into moment contributions, and depositing those contributions in destination cells. The streamed contributions are

$(D+K)$1

and the landing-cell update is

$(D+K)$2

Streaming and moment transformation are therefore fused into a single operation (Noh et al., 9 Aug 2025).

The one-step workflow described in the paper is local except for the destination update during streaming. A cellwise step consists of reading stored moments, computing $(D+K)$3, $(D+K)$4, and $(D+K)$5, transforming to central moments, applying the collision map with $(D+K)$6, optionally applying a Knudsen limiter, building the local LAS, computing relative velocities, performing EPR separately for $(D+K)$7 and $(D+K)$8, and streaming the resulting moment fractions. The update remains explicit because collision, LAS construction, and EPR are all local, and no global linear system is solved (Noh et al., 9 Aug 2025).

The paper states that the dominant computational cost lies in EPR. For a stencil of size $(D+K)$9, each Newton iteration costs approximately

$D$0

plus

$D$1

for the small linear solves. Since $D$2, matrix assembly is reported to dominate. In a “hot-box” test, the scaling variable is given as

$D$3

and the measured wall-clock time is reported to scale linearly with this quantity. On an RTX 4070 GPU, the paper reports approximately 10.9 s/timestep at $D$4, 33.8 s at $D$5, and 408.5 s at $D$6 for a $D$7 problem. On GPU, atomic accumulation is implemented with atomicAdd in CUDA (Noh et al., 9 Aug 2025).

A plausible implication is that LELBM trades the low arithmetic intensity of standard stream-collide LBM for a higher local reconstruction cost in exchange for adaptive velocity support and constant per-cell memory. That trade-off is explicit in the reported design, although the paper does not reframe it in those exact terms.

6. Stability mechanisms, validated regimes, and limitations

The paper attributes LELBM’s stability to five interacting mechanisms: central-moment collision, entropy-based reconstruction, adaptive LAS support, a Knudsen-number-based limiter, and bulk-mode damping. Central moments improve numerical precision by separating large bulk motion from non-equilibrium content. EPR prevents negative populations while enforcing prescribed moments. LAS matches local velocity support to local thermodynamic state. The Knudsen limiter suppresses extreme non-equilibrium moments near shocks. The paper also notes that $D$8 is commonly used to suppress oscillatory acoustic artifacts (Noh et al., 9 Aug 2025).

The local Knudsen estimate used by the limiter is

$D$9

When this exceeds continuum thresholds, collision frequencies are increased toward $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$00 so that the state is pulled closer to equilibrium and reconstruction becomes more likely to succeed (Noh et al., 9 Aug 2025).

The validated physical-property tests include shear wave, thermal wave, acoustic plane wave, acoustic pulse, and hot-box advection. In the shear-wave test, the paper reports good viscosity accuracy near $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$01, activation of internal velocity refinement around $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$02, numerical viscosity from refinement of approximately $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$03, and increased numerical diffusion at higher $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$04. In the thermal-wave test, the paper reports best accuracy for $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$05, excessive diffusion at low $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$06 when refinement is active, and recommends $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$07 for thermoviscous problems. In the acoustic pulse test, the method is reported to match

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$08

across $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$09 to 100 and for $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$10, corresponding to $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$11 (Noh et al., 9 Aug 2025).

The principal shock and flow benchmarks are Sod’s shock tube, the Lax problem, Shu–Osher, a 2D Riemann problem, double Mach reflection, oblique shocks, supersonic flow past a circular cylinder, and supersonic flow past a NACA0012 airfoil. The paper reports excellent agreement with the exact solution for Sod’s problem; robust accuracy in the hotter and acoustically rescaled variants of the Lax problem; well-resolved downstream oscillatory structures in Shu–Osher; excellent agreement with reference solutions for the 2D Riemann problem; resolved primary shock, Mach stem, and slip line in double Mach reflection; excellent agreement with analytical $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$12-$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$13-Ma relations for oblique shocks; good shock-stand-off behavior for the cylinder case; and very good pressure coefficient agreement for the NACA0012 airfoil at $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$14 (Noh et al., 9 Aug 2025).

The paper’s stated accessible regime includes sound-speed validation up to $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$15, practical accuracy best near $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$16, oblique shocks up to $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$17, cylinder flow up to $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$18, and lattice Courant numbers exceeding 30. It specifically reports an LAS example with $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$19, oblique shock up to 32.7, and notes that larger values are possible. The minimum temperature in the acoustically scaled double-Mach-reflection setup is reported as 0.179, which sits close to the stated low-temperature difficulty threshold (Noh et al., 9 Aug 2025).

The limitations are also stated explicitly. The most significant is low-temperature robustness: EPR begins to struggle near $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$20 without refinement, refinement helps only down to roughly $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$21, and failures still occur near $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$22. Bulk-viscosity control is less accurate than shear and thermal diffusivity, particularly at high acoustic frequencies. The current implementation lacks higher-order time integration, flux control or high-order shock-capturing corrections, and mesh refinement. Curved bodies are represented by a Cartesian “legoland” approximation, the boundary treatment is mainly for adiabatic walls, temperature-specified wall boundaries are said not to be generally accurate in the current form, and most demonstrations are one- and two-dimensional. The paper further identifies generalized boundaries, unstructured meshes, three-dimensional practical extension, chemistry, and high-fidelity turbulent supersonic applications as future work (Noh et al., 9 Aug 2025).

These limitations distinguish LELBM from entropic methods developed for other purposes. Earlier entropic work established stability mechanisms for equilibrium construction, moving and deforming geometry treatment, and multilevel coupling, but did not solve the low-temperature adaptive-stencil reconstruction problem that is specific to this Lagrangian compressible framework. This suggests that LELBM’s open issues are tied less to entropy minimization itself than to the interaction of adaptive local stencils, moment reconstruction, and strong compressibility (Dorschner et al., 2016).

7. Relation to the broader entropic lattice Boltzmann literature

LELBM occupies a specific position within the entropic lattice Boltzmann family. Earlier entropic work provided several components that are conceptually adjacent but not equivalent. Entropic equilibria on tensor-product lattices were already known in factorized product form, and their origin from discrete entropy minimization was analyzed in both theoretical and kinetic contexts. The entropic equilibrium on D2Q9, for example, had been connected to minimization of

$$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$23

and Monte Carlo lattice-gas constructions had shown how detailed-balance collisions recover the same equilibrium in the Boltzmann average (Blommel et al., 2017).

Other works had clarified different roles of entropy. One line of research derived analytical approximations for the entropic path-length parameter $$\mathrm{Co} = (|u| + c_s)\frac{\Delta t}{\Delta x},$$24 and reformulated ELBM as an “essentially entropic” collision method that enforces entropy decrease without iterative involution. Another analyzed the entropic equilibrium itself and argued that its velocity-dependent pressure corrections produce unconditional linear stability and positive-definite dissipation over the admissible lattice-speed interval. These results established entropy minimization and entropy-monotone collision as stability devices, but they remained attached to Eulerian fixed-lattice streaming (Hosseini et al., 2023).

Entropic techniques had also been extended to moving and deforming geometries on fixed Cartesian grids. Three-dimensional KBC entropic methods with Grad-based moving-wall reconstruction, node refill, and two-way fluid–structure interaction were demonstrated for a sedimenting sphere, a plunging airfoil, and an anguilliform swimmer. Likewise, entropic grid-refinement strategies were developed for isothermal, thermal, and compressible models, with explicit scaling rules for nonequilibrium transfer across levels. These works are relevant background for LELBM because they show that entropy-based collision and stabilization are compatible with moving boundaries and multiscale coupling, but they are not Lagrangian formulations in the sense used by LELBM (Dorschner et al., 2016).

The 2025 analytical-solution paper on entropic equilibria is even more narrowly related. It derives arbitrary-order derivatives of the factorized entropic equilibrium for Couette-flow calculations, using Faà di Bruno’s formula and Bell-polynomial identities, and it explicitly states that it does not discuss anything called “Lagrangian Entropic Lattice Boltzmann Method.” Its relevance to LELBM is therefore indirect: it highlights the nonlinear derivative structure of entropic equilibria that any Lagrangian or semi-Lagrangian entropic method would also inherit (Larson et al., 30 May 2025).

Against this background, LELBM is distinguished by the specific combination of five elements in a single explicit compressible method: regularized central moment collision, polyatomic DDF modeling, adaptive shifted multispeed LAS, entropic moment-constrained reconstruction in the relative frame, and moment-streaming storage compression. The paper presents this combination as the first fully unified framework of that kind for explicit, positivity-preserving, memory-efficient supersonic lattice Boltzmann simulation (Noh et al., 9 Aug 2025).