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Mesh-Uniform Power Stability of Two-Relaxation-Time Vector Lattice Boltzmann Schemes with Reversible Boundaries

Jin Zhao

arXiv:2608.30253v1math.NA

TL;DR

The paper asks whether rest-state linearized vector TRT schemes remain uniformly power stable under reversible boundary transport and whether convex off-midpoint interpolation guarantees stability. It proves a contraction-based, dimension-independent bound using a two-step recurrence, numerical-range ellipses, and Crouzeix–Palencia, while exhibiting an admissible D2N5 counterexample to unconditional convex-coefficient stability.

  • Problem

    Boundary-closed lattice Boltzmann operators require stability analysis beyond periodic interiors because boundary operators may create unstable modes or nonnormal growth.

  • Method

    The proof converts the collision–transport update into a two-step macroscopic recurrence generated by a contraction, then uses numerical-range control and the Crouzeix–Palencia theorem.

  • Results

    The amplification powers are uniformly bounded over finite lattices and reversible transports, while an exact rational D2N5 example has a real eigenvalue larger than one for admissible off-midpoint data.

  • Takeaways & Limitations

    Coefficient nonnegativity alone is not an unconditional stability condition for the three-coefficient off-midpoint boundary formula.

  • Takeaways & Limitations

    The analysis does not provide a full boundary-trace estimate for arbitrary incoming data, and its separable orthogonalizer obstruction does not imply instability of the complete product.

Abstract

from arXiv · show

We consider the collision-transport operator for vector-valued two-relaxation-time lattice Boltzmann schemes linearized about a uniform rest state. Assume that the equilibrium blocks are positive definite and that the link-even and link-odd relaxation parameters satisfy s_+ + s_- = 2 and 0 < s_- < 2. If the homogeneous transport is unitary in the equilibrium metric and reversible under velocity exchange, then the powers of the amplification operator are bounded uniformly with respect to the number and arrangement of lattice nodes. The admissible transports include periodic transport, vector halfway bounce-back, coordinate-aligned specular reflection, tangential orthogonal involutions, and compatible multi-channel scattering. The proof reduces the population equation to a two-step macroscopic recurrence generated by a contraction. An inclusion of the numerical range in an ellipse, combined with the Crouzeix-Palencia theorem, yields a dimension-independent estimate for the companion operator and hence the population bound. For a three-coefficient off-midpoint boundary interpolation, we give an exact rational D2N5 example whose finite-domain amplification matrix has a real eigenvalue larger than one, although the interpolation coefficients and bulk parameters are admissible. Thus coefficient convexity alone does not ensure stability for this boundary family.

1 Introduction

The paper studies mesh-uniform power stability for rest-state linearized vector TRT lattice Boltzmann schemes with reversible boundary transport, while identifying limits of separable energy arguments and off-midpoint interpolation.

  • Boundary closure requires separate stability analysis because it can introduce unstable modes or substantial nonnormal growth despite stable periodic interiors.
  • The paper analyzes homogeneous reversible scattering for complete vector population states and proves a mesh-uniform bound for the rest-state linearization.
  • The equilibrium metric makes lifting isometric and reversible transport unitary, reducing the population update to a macroscopic contraction-based recurrence.
  • Numerical-range control and the Crouzeix–Palencia theorem yield dimension-independent companion and population estimates for periodic, bounce-back, specular, involutive, and compatible scattering transports.
  • No block-diagonal positive-definite symmetrizer simultaneously orthogonalizes equilibrium projection and commutes with velocity reversal when the lattice-speed parameter is nonzero.
  • An exact rational D2N5 example has a real amplification eigenvalue larger than one with positive interpolation coefficients and admissible bulk parameters, so convexity alone is insufficient.

3 Reversible boundary transport

The section defines reversible, equilibrium-compatible transports that are unitary in the equilibrium metric and time-reversible under velocity exchange. It verifies the class for linkwise reflections, periodic links, tangential involutions, and compatible multi-channel boundary scattering.

  • Equilibrium-compatible transport: The equilibrium metric makes the lifting isometric, while compatible transport is unitary and reversible under velocity exchange.These properties support the macroscopic compression and its adjoint identity.
  • Linkwise criterion: The global transport criterion requires paired source-target links with inverse component maps under velocity reversal.Under these conditions, the transport satisfies T †T = I and T −1 = J T J.
  • Multi-channel scattering: Patchwise orthogonal mixing yields a compatible multi-channel wall-scattering class, but no off-midpoint accuracy claim is attached.The construction provides homogeneous stability compatibility rather than an accuracy result.
  • Realized transports: Periodic transport, vector halfway bounce-back, coordinate-aligned specular reflection, and mixtures of these links satisfy the reversible transport class.The criterion is not restricted to rectangular domains.
  • Realized transports: Tangential orthogonal involutions extend the reflection class, with no-slip and specular reflection as endpoint choices.In three dimensions, other orthogonal involutions of the tangential plane provide additional component reflections.
  • Macroscopic compression: The resulting macroscopic compression P is a contraction in the Euclidean macroscopic norm.This follows from equilibrium lifting and transport unitarity.

4 A mesh-uniform companion estimate

The section establishes a dimension-independent power bound for the companion recurrence generated by an arbitrary contraction. Numerical-range inclusion in an ellipse and the Crouzeix–Palencia theorem transfer scalar polynomial bounds to the nonnormal companion operator.

  • Companion recurrence: The companion recurrence is analyzed for an arbitrary contraction on a finite-dimensional complex Hilbert space.The resulting estimate is designed to remain uniform as the Hilbert-space dimension changes.
  • Numerical-range ellipse: For a contraction P, the numerical range of P + bP ∗ lies in the ellipse E_b = {ζ + bζ : |ζ| ≤1}.The inclusion is the geometric input for the operator estimate.
  • Scalar recurrence: The scalar recurrence polynomials q_n satisfy q0(s) = 0, q1(s) = 1, and qn+1(s) = sqn(s) − bqn−1(s).Their uniform bound is first established on the ellipse using its boundary parametrization and the maximum-modulus principle.
  • Operator estimate: The Crouzeix–Palencia theorem transfers the polynomial bound from the numerical range to the nonnormal companion operator.This produces a uniform companion estimate without requiring normality.
  • Uniformity and limitation: The companion bound is independent of the dimension and the particular contraction.Its constant is not claimed sharp and deteriorates as |b| approaches one, consistently with loss of damping at relaxation endpoints.

5 Mesh-uniform power and resolvent stability

Under the OTRT relation, reversible metric-compatible transports yield mesh-uniform power, resolvent, and weighted ℓ2 bounds through a contractive macroscopic recurrence.

  • Macroscopic reduction: The proof uses an exact two-step macroscopic recurrence whose governing compression is a contraction.The equilibrium lifting is isometric, homogeneous transport is unitary, and reversibility supplies the recurrence estimates.
  • Uniform power stability: The complete linearized amplification operator is power bounded uniformly over finite lattices and admissible reversible transports.The explicit bound is independent of the number and arrangement of lattice nodes.
  • Assumptions: The theorem applies under 0 < α < 2a < 1/d, 0 < s_- < 2, and s_+ = 2 − s_- .These conditions ensure positivity and the OTRT rate-sum relation used by the stability argument.
  • Mesh refinement: Uniform estimates extend across arbitrary refining lattice families and grid-weighted ℓ2 norms.Common cell-volume factors do not change the induced operator estimates.
  • Reversible boundaries: The admissible boundary class includes halfway bounce-back, coordinate-aligned specular reflection, tangential orthogonal involutions, and compatible patchwise mixing.The stability constant remains independent of node count and arrangement.
  • Resolvent and scope: The results also provide uniform exterior resolvent and exponentially weighted ℓ2 estimates, but not a GKS strong-stability theorem.Sharp boundary-trace estimates for arbitrary incoming data and off-midpoint boundary symbols require additional analysis.

6 The off-midpoint parameterized boundary

The three-coefficient off-midpoint boundary rule generally falls outside the reversible-transport factorization, and an exact admissible D2N5 example demonstrates instability despite convex coefficients.

  • Boundary structure: The interpolation formula mixes pre- and postcollision populations except on the degeneracy line ℓ = 2γ − 1.Only the halfway case (γ, ℓ) = (1/2, 0) is directly covered by the reversible postcollision transport class.
  • Postcollision-only case: On a forward-fluid-neighbor link, D-unitarity forces the postcollision-only subfamily to reduce to vector halfway bounce-back.This conclusion assumes ordinary forward streaming and a transport on the original population space.
  • Exact counterexample: The exact rational D2N5 construction uses positive coefficients 1/2, 1/4, and 1/4 with admissible bulk parameters, yet its full 135 × 135 amplification matrix has a real eigenvalue larger than one.The instability is established through an invariant symmetry subspace and exact rational characteristic-factor calculations.
  • Interpretation: The counterexample excludes the full convex-coefficient range as an unconditional stability region, not every off-midpoint parameter.Finite-domain sweeps contain both spectrally unstable values and values with no detected eigenvalue outside the unit disk.
  • Bulk-versus-boundary comparison: The comparison with halfway bounce-back indicates that the unstable eigenvalue is not caused by the bulk OTRT collision alone.For the same domain and collision parameters, halfway bounce-back has spectral radius one to numerical precision.

7 Further questions in boundary stability

Further progress requires distinguishing reversible scattering rules from accuracy-oriented pre/postcollision closures and analyzing off-midpoint boundaries with boundary-specific tools or augmented energy structures.

  • Reversible boundary class: The theorem extends beyond bounce-back to bijective weighted-unitary scattering reversed by velocity exchange.This includes linkwise permutations, tangential orthogonal involutions, and compatible multi-channel scattering.
  • Coverage boundary: Accuracy-oriented boundary frameworks are covered only when their complete homogeneous scattering map satisfies the theorem’s reversibility criterion.Neighboring-node, memory, and pre/postcollision rules generally require an augmented energy or joint collision-boundary estimate.
  • Boundary-symbol analysis: A stable off-midpoint subset should be tested through tangential Fourier reduction and a wall-normal boundary-symbol analysis.The analysis must include generalized eigenvalues on and outside the unit circle, glancing modes, and boundary-bulk resonances.
  • Nonnormality: For nonnormal off-midpoint operators, spectral radii alone are insufficient; pseudospectra and powers should also be examined.A mesh-independent theorem additionally requires uniformity in tangential wave number and wall-normal resolution.
  • Energy-compatible extension: An alternative is to add finite boundary memory and impose passivity, reversal, and consistency identities in an extended metric.The paper states algebraic constraints for such a rule but does not construct one.

8 Conclusion

The paper proves mesh-uniform stability for reversible OTRT boundaries while showing that convex off-midpoint interpolation coefficients do not guarantee stability.

  • Main stability result: The OTRT rate-sum relation converts the update into a two-step recurrence whose contraction-based analysis yields population-space power bounds.Numerical-range control of the companion recurrence makes the estimate uniform over finite lattices and reversible transports.
  • Covered transports: The reversible class covers periodic transport, halfway bounce-back, coordinate-aligned specular reflection, tangential orthogonal involutions, and compatible multi-channel scattering.These are the concrete transport families included by the theorem.
  • Boundary limitation: The exact D2N5 example shows that nonnegative interpolation coefficients are not an unconditional stability condition for the studied off-midpoint boundary family.Open problems include identifying a restricted stable set and constructing an energy-compatible augmented boundary rule.

Computational verification

The instability conclusion is supported by exact rational identities and sign evaluations, with floating-point calculations limited to reported decimals and illustrative experiments.

  • Computational verification: Exact rational identities and sign evaluations establish the instability assertion, while floating-point computation supplies only reported decimals and illustrative numerical experiments.The supplementary materials document the matrix conventions, invariant-subspace calculation, and auxiliary finite-domain spectral computations.

Supplementary Materials

The supplementary materials document numerical matrix assembly, the exact rational calculation behind Proposition 6.2, and auxiliary finite-domain computations, while the main article contains the analytical stability proof.

  • Supplementary Materials: The paper title identifies mesh-uniform power stability for two-relaxation-time vector lattice Boltzmann schemes with reversible boundaries.The title spans the two supplied title passages.
  • Supplementary Materials: The supplement specifies matrix assembly, gives the exact rational calculation supporting Proposition 6.2, and reports auxiliary finite-domain computations.It distinguishes these materials from the analytically proved mesh-uniform stability theorem in the main article.

Matrix assembly

The matrix assembly uses node-major, velocity-major, component-major ordering, local lifting and reversal blocks, global Kronecker constructions, and boundary-specific source rules.

  • Matrix assembly: Population blocks use velocity ordering (+e1, …, +ed, −e1, …, −ed, 0) and component order (ρ, m1, …, md).The global vector is ordered node-major, then velocity-major, then component-major.
  • Matrix assembly: Local lifting, moment, and reversal matrices are assembled from Kloc, Sloc, and opposite-velocity identity blocks, with global matrices formed using IN.The reversal matrix places an Im block at the position associated with each opposite velocity.
  • Matrix assembly: The equilibrium weight used in the main article is included as part of the matrix-assembly specification.The supplied passage identifies the weight but does not reproduce its formula.
  • Boundary rules: Linkwise reversible rules assign each target population one source, translating interior sources with Im and closing missing links using R0 or coordinate reflections Rj.The supplied rule uses R0 = diag(1, −Id) or Rj = diag(1, Id − 2ejeT).
  • Compatible scattering: The larger compatible scattering class mixes several outgoing source blocks into equal numbers of incoming targets subject to two exact matrix identities.Numerical residuals verify matrix assembly but are not used in the analytical proof.
  • Parameterized boundary: The parameterized boundary matrix uses block-diagonal collision, and missing-link target rows receive the stated interpolation-based incoming expression.The convention agrees with the incoming-direction formula in the main article.

Exact rational counterexample

The D2N5 counterexample reduces the rational amplification matrix to an invariant symmetry subspace and proves an eigenvalue above one through exact characteristic-factor sign changes.

  • Symmetry reduction: The 135-dimensional D2N5 amplification matrix commutes with quarter-turn rotation and vertical reflection acting on nodes, velocities, and momentum components.All parameters and entries are rational, enabling an exact symmetry reduction.
  • Symmetry reduction: The group projector has rank 18, and independent integer columns provide a basis for the selected symmetry type.The supplied passages identify the projector as ΠB/8 and state its rank.
  • Invariant restriction: The exact identity AB = BM proves invariance of the basis range and identifies M as the restricted operator whose characteristic polynomial contains p9.The rational factorization and verification data are supplied with the construction.
  • Instability proof: A real zero of p9 lies in (201/200, 503/500), so the full amplification matrix has an eigenvalue exceeding one.The conclusion follows from opposite exact signs and the intermediate value theorem, without floating-point eigenvalue computation.

Numerical verification of structural identities

Numerical checks verify the reversible structural identities, recurrence relations, and companion estimates, while finite-domain calculations illustrate transient growth and domain dependence without establishing mesh-independent stability.

  • Structural identities: Residuals below 3 × 10−15 verify the six tested reversible identities in representative OTRT calculations.The checks include D3N7 and one- and two-dimensional cases with specified admissible parameters.
  • Recurrence verification: The D2N5 L-shaped test has relative residuals below 3 × 10−16 for both the macroscopic recurrence and the population-moment identity.The computed norm of P is one to displayed precision.
  • Boundary extensions: The full-vector boundary extension is also verified exactly in D3N7 using rational tangential and patch-channel involutions.This calculation uses a = α = 1/7.
  • Companion estimates: For random nonnormal contractions and truncated unilateral shifts, observed power norms remain below the explicit companion-theorem constant.The largest observed ratio to the nonsharp constant is approximately 0.208, with block-identity residual below 2 × 10−15.
  • Finite-domain calculations: Finite-domain spectral plots record parameter- and size-dependent behavior, but sampled unit-disk spectra do not prove power stability or define a mesh-independent stability region.The exact counterexample is marked at (γ, ℓ, η) = (3/4, 1, 1/2), while strip values for N = 4, . . . , 9 differ from one by at most 1.1 × 10−14.
  • Finite-domain calculations: The off-midpoint counterexample reaches approximately 42.8 in computed power norm at step 300, versus approximately 2.216 for halfway bounce-back.These are Euclidean norms of powers on the 3 × 3 square.
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