Lagrangian Acoustic Stencils (LAS)

• Lagrangian Acoustic Stencils (LAS) are adaptive discretization schemes that build local velocity stencils based on material motion and thermal scales.
• LAS integrate approaches from LELBM, variational finite elements, and transported-acoustic increments to ensure high-order accuracy and energy-consistent reconstruction.
• The strategy unifies multiple design principles, offering Courant-free, explicit simulations for compressible flows and shock-dominated phenomena.

Lagrangian Acoustic Stencils (LAS) are acoustic discretizations formulated in a co-moving, material, or locally shifted frame rather than on a globally fixed stencil. In the most explicit use of the term, LAS are introduced as a generalized multispeed shifted lattice construction method within the Lagrangian entropic lattice Boltzmann method (LELBM): in each cell and at each time step, a local velocity stencil is centered on the local bulk velocity and scaled by the local thermal speed, then used transiently for entropic reconstruction and moment streaming (Noh et al., 9 Aug 2025). In a broader technical sense suggested by adjacent work, LAS also denote a family of Lagrangian or Lagrangian-like acoustic operators built from compact multidimensional stencils, variational couplings, transported acoustic increments, or material-frame boundary formulations (Fu et al., 2023, Nama et al., 2016, Duraisamy, 13 May 2026).

1. Terminology, scope, and research setting

The strictest definition of LAS comes from the LELBM literature, where the stencil is a local velocity-space object adapted to arbitrary local velocity and temperature (Noh et al., 9 Aug 2025). That usage is specific: LAS are not merely spatial finite-difference templates, but locally rebuilt multispeed shifted lattices that support explicit, Courant-free, supersonic-capable compressible-flow simulation.

At the same time, several earlier or parallel lines of work supply the structural ingredients now associated with LAS. High-order variational Lagrangian finite elements provide acoustically relevant stencils through local FE couplings derived from discrete energy and dissipation functionals (Fu et al., 2023). ALE acoustic streaming formulations place the first-order acoustic field in terms of displacement and the second-order problem directly in terms of Lagrangian mean velocity, with exact wall conditions on the mean configuration (Nama et al., 2016). Active Flux formulations for compressible Euler preserve a compact exact acoustic evolution operator and transport its increment to the convective foot, yielding a materially transported acoustic correction (Duraisamy, 13 May 2026).

Two commonly associated papers are not LAS works in the strict terminological sense. The convolutional dispersion-relation-preserving (DRP) stencil of the acoustic wave equation is explicitly described as “strictly Eulerian and finite-difference-based” and “does not mention Lagrangian, moving grids, or particle methods explicitly” (Ovadia et al., 2022). Likewise, the Poisson-formula stencil schemes are “entirely Eulerian” and derived on a fixed Cartesian grid (Khutoryansky, 2019). Their importance for LAS lies in transferrable design principles rather than direct nomenclature.

2. LAS as adaptive velocity-space stencils in entropic LBM

In LELBM, LAS are designed around the continuous Maxwellian

$$f^{\mathrm{eq}}(\rho,\boldsymbol{\xi},\mathbf{u},RT)=\frac{\rho}{(2\pi RT)^{D/2}}\exp\left(-\frac{|\boldsymbol{\xi}-\mathbf{u}|^2}{2RT}\right),$$

whose mean is the local bulk velocity $\mathbf{u}$ and whose width is controlled by $\sqrt{RT}$ (Noh et al., 9 Aug 2025). LAS construct a discrete velocity set in the relative frame $\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$ so that the local stencil tracks both convection and acoustic scale.

Three diagnostics organize this construction. The conformation number is

$$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$

the lattice Courant number is

$$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$

and the interlattice Courant number is

$$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$

The key LAS constraint defines an acoustic radius in relative velocity space:

$$|\mathbf{c}_i-[\mathbf{u}]|\Delta t\le R_{\mathrm{ac}}=\max\Big([n\sqrt{RT}\,\Delta t],\,2\Delta x\Big).$$

The reported choice is $n=4$, which captures $\approx 99.99\%$ of a Gaussian. The lower bound $\mathbf{u}$0 guarantees at least 5 stencil points along any direction, which the paper states is enough to reconstruct up to third-order moments isotropically (Noh et al., 9 Aug 2025).

The method distinguishes high- and low-temperature regimes. For high temperature, the number of velocities scales as

$\mathbf{u}$1

which is the usual polynomial growth of multispeed LBM. For low $\mathbf{u}$2, the Gaussian peak becomes too narrow for a uniform integer velocity spacing, so LAS introduce an internal refinement with

$\mathbf{u}$3

and nested refinement regions

$\mathbf{u}$4

with velocity volumes scaled by

$\mathbf{u}$5

Even with refinement, the reported entropic reconstruction becomes unreliable and very diffusive below about $\mathbf{u}$6 (Noh et al., 9 Aug 2025).

Algorithmically, LAS occupy one module in a four-part cycle consisting of collision, LAS construction, entropic population reconstruction (EPR), and moment streaming (MS). EPR reconstructs normalized post-collision populations from central-moment constraints using a discrete entropy functional

$\mathbf{u}$7

and LAS determine the relative velocities entering those constraints. MS then streams moment contributions rather than full populations, so persistent storage remains fixed. In 3D, the paper states that only 17 moments per cell are stored, namely 13 fluon and 4 phonon moments, independent of $\mathbf{u}$8 (Noh et al., 9 Aug 2025).

This LAS construction is the mechanism behind the paper’s “Courant-free” claim. Stability is tied not to a conventional continuum CFL restriction but to conformation, positivity, and reconstruction quality. The method is validated on Sod’s shock tube, the Lax problem, Shu–Osher waves, 2D Riemann problems, double Mach reflection, oblique shocks, supersonic flow past a circular cylinder, and supersonic flow past a NACA0012 airfoil. Reported operating regimes include unscaled Lax with maximum $\mathbf{u}$9 and $\sqrt{RT}$0, double Mach reflection with $\sqrt{RT}$1, and oblique shocks with $\sqrt{RT}$2 up to 20 and maximum local $\sqrt{RT}$3 (Noh et al., 9 Aug 2025).

3. Transferable stencil-design principles from DRP and Poisson-based acoustics

Although not Lagrangian papers, two Eulerian acoustic-stencil lines supply reusable LAS design logic. The first is the convolutional DRP scheme for the 2D acoustic wave equation. Its update is

$\sqrt{RT}$4

with a symmetric $\sqrt{RT}$5 stencil parameterized by six weights. The parameterization enforces rotational and reflection symmetry, and analytic constraints impose zero-sum consistency, second-order matching of the Laplacian, and optionally fourth-order cancellation of Taylor terms. The machine-learning component then optimizes within that constrained family using a one-step MSE and a three-step iterative loss that penalizes accumulated phase error over unrolled time integration (Ovadia et al., 2022).

The supplied technical analysis identifies the LAS analogue explicitly: parameterize an inter-particle kernel or neighbor stencil, enforce zeroth-, second-, and higher-moment constraints to guarantee consistency and polynomial accuracy, and train the remaining free parameters with a multi-step loss so that the discrete dispersion relation matches the physical relation $\sqrt{RT}$6 over a band of wavenumbers (Ovadia et al., 2022). This is not a claim that the paper itself is Lagrangian; it is a transfer of its scheme-design strategy.

The second line is the Poisson-formula construction of explicit 2D wave stencils. There the exact three-time-level integral evolution is approximated by local polynomial interpolation and exact integration, leading to compact 5-, 9-, and 13-point explicit schemes. The 5-point two-step update coincides with the standard second-order scheme, but the first-step formula includes an additional spatial-difference term in the initial velocity and removes an $\sqrt{RT}$7 start-up error. The paper reports stability bounds $\sqrt{RT}$8 for the 5-point scheme, $\sqrt{RT}$9 for the new 9-point scheme, and the same $\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$0 limit for the 13-point fourth-order scheme (Khutoryansky, 2019).

For LAS, the explicit relevance is the combination of local reconstruction and exact or high-order evolution. The supplied analysis states that this is “exactly the design pattern” one wants for moving Lagrangian stencils, except that the fixed Cartesian stencil is replaced by a moving particle neighborhood (Khutoryansky, 2019). A plausible implication is that two LAS design motifs recur across otherwise different discretizations: constrained local approximation of the acoustic operator, and evolution formulas that remain consistent from initialization through subsequent steps.

4. Variational Lagrangian stencils and energetic structure

High-order variational Lagrangian finite elements provide a different LAS lineage, one grounded in discrete energetics rather than in fixed stencils. For compressible fluids, the method derives the governing equations from a discrete energetic variational approach. Positions and velocities live in a continuous FE space

$\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$1

while thermodynamic variables live in a discontinuous quadrature-point space

$\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$2

The discrete action is built from kinetic and free energies, and force balance is imposed as a discrete variational identity (Fu et al., 2023).

The resulting semi-discrete velocity equation is

$\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$3

with nodal trajectory equation $\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$4 and a quadrature-point temperature ODE. In this setting, the “stencil” is the FE connectivity induced by local support and quadrature, so acoustic coupling is expressed through a pressure-divergence operator and a velocity-divergence operator assembled from local element contributions (Fu et al., 2023).

Three discrete structural properties are central. Mass conservation is exact at quadrature points through

$\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$5

Momentum is conserved semi-discretely. Total energy satisfies

$\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$6

and entropy obeys

$\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$7

when $\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$8 and viscosities are non-negative (Fu et al., 2023). In LAS terms, these identities define an energetically consistent stencil architecture: the acoustic operator is not chosen ad hoc, but obtained as the variational derivative of discrete energy and dissipation.

Time integration is fully implicit. The paper develops backward Euler, a midpoint-rule variant that is energy-conservative at the fully discrete level, and high-order BDF schemes. It reports that, in low-Mach regimes, time steps can be “two to three orders of magnitude larger” than explicit methods. In the low-Mach Gresho vortex, degree $\tilde{\mathbf{c}}_i=\mathbf{c}_i-\mathbf{u}$9 shows locking as Mach decreases, while degree $$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$0 gives nearly identical results for $$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$1 (Fu et al., 2023). This identifies high-order spatial approximation and implicit variational coupling as key ingredients for low-Mach LAS.

5. Material transport of acoustic information

A distinctive LAS feature is that acoustic information is naturally tied to material motion. In the ALE formulation of acoustic streaming, the fast first-order problem is written in terms of displacement $$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$2 on a mean configuration, with first-order velocity

$$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$3

The second-order problem, after time averaging and scale separation with $$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$4, becomes a steady Stokes-type system for the Lagrangian mean velocity

$$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$5

The formulation solves directly for $$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$6, so no separate Stokes-drift post-processing is required, and the exact streaming boundary condition on the mean configuration is

$$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$7

(Nama et al., 2016). In LAS terms, the first-order stencil acts on displacement, while the second-order stencil is a steady Stokes operator driven by quadratic combinations of first-order displacement and its gradients.

The transported-acoustic-increment Active Flux method gives an even more explicit materially transported LAS construction. It starts from the frozen linearized acoustic subsystem

$$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$8

with acoustic increment

$$n=\frac{\big|\big|\overline{|\mathbf{c}-\mathbf{u}|}-R_{\mathrm{ac}}\big|\big|}{\sqrt{RT}},$$9

Those increments are reconstructed cellwise as a $$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$0 field and evaluated at the convective foot

$$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$1

leading to the point update

$$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$2

Under constant frozen coefficients, this transported update reduces to

$$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$3

which equals the exact unsplit operator $$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$4 when the advective and acoustic generators commute (Duraisamy, 13 May 2026).

The paper interprets Barsukow’s exact compact acoustic operator as a LAS-like object and preserves it unchanged; the novelty is that its increment is reattached to the advected material label rather than added at a fixed Eulerian node. Numerically, the transported update yields third-order point accuracy for a mixed Fourier wave packet, compared with second-order behavior for the additive update, and shows improved acoustic phase and amplification behavior in Fourier diagnostics (Duraisamy, 13 May 2026). This supports a general LAS principle: acoustic corrections should be transported with advection when the governing operator is materially convected.

6. Boundaries, stability, and present limitations

The most explicit Lagrangian boundary formulation appears in the study of acoustic waves interacting with non-locally reacting surfaces. The bulk unknown is the fluid-particle displacement $$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$5, and the moving boundary carries a normal displacement $$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$6. The interior equations are

$$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$7

while the moving boundary satisfies

$$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$8

with kinematic coupling

$$\mathrm{Co}_{\mathrm{lat}}=\max_i |\mathbf{c}_i|\frac{\Delta t}{\Delta V^{1/D}},$$9

Here the non-local reaction is spatial, not temporal: neighboring points on $$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$0 interact through the surface operator $$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$1 (Vitillaro, 8 Jan 2025).

The associated energy is

$$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$2

and it satisfies

$$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$3

The paper proves unique weak solvability in the natural Hilbert spaces and asymptotic stability when $$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$4 on $$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$5 and $$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$6 (Vitillaro, 8 Jan 2025). For LAS, this fixes the continuous target for any discrete treatment of non-locally reacting boundaries: compatibility of trace operators, surface Laplace–Beltrami discretization, and a discrete energy identity are not optional implementation details but structural requirements.

Current limitations differ by LAS lineage. In LELBM, low-temperature entropic reconstruction remains problematic below about $$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$7, and robust boundary support is limited primarily to adiabatic walls (Noh et al., 9 Aug 2025). In transported-acoustic-increment Active Flux, the strongest analysis is the frozen linear setting, the numerical study is restricted to no-limiter smooth-flow and under-resolved tests rather than shocks, and exact off-center Mach-disk integration is reported to be about $$\mathrm{Co}_{\mathrm{int}}=\big|\overline{\mathbf{c}-\mathbf{u}}\big|\frac{\Delta t}{\Delta V^{1/D}}.$$8 slower than the transported-increment alternative (Duraisamy, 13 May 2026). In the variational Lagrangian FE line, mesh regularization or remapping is not included, and extension to ALE is identified as future work (Fu et al., 2023). Meanwhile, the convolutional DRP and Poisson-formula stencil papers remain Eulerian sources of transferable ideas rather than LAS implementations proper (Ovadia et al., 2022, Khutoryansky, 2019).

This indicates that LAS currently name a family of related constructions rather than a single standardized discretization. What unifies them is not one algebraic form, but a common insistence that acoustic propagation should be represented on stencils tied to local material state, local acoustic scale, or materially transported increments, while preserving consistency, multidimensional propagation, and—where the formulation permits—energy or entropy structure.