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Sparse entropic quadrature for moment equations

Georgii Oblapenko, Michael Herty, Manuel Torrilhon

arXiv:2609.29843v1math.NAcs.NA

TL;DR

Rarefied gas moment equations need closure because their fluxes depend on unknown higher-order moments, and closure methods must also address properties such as hyperbolicity and realizability. This paper analyzes sparse entropic quadrature, using constrained optimization with a weighting function and sparsity in the reconstructed distribution. It establishes closure properties and reports accurate simple test-case solutions with little accuracy impact from substantial sparsity.

  • Problem

    Moment closure must estimate unknown higher-order moments while addressing non-negativity, hyperbolicity, conditioning, and computational efficiency.

    Source: §1 · p. 2

  • Method

    Sparse entropic quadrature reconstructs the distribution by constrained optimization, decomposing it into a weighting function and a sparse unknown component.

    Source: §3 · p. 5

  • Results

    The closure has existence, uniqueness, and smoothness properties; it is strictly hyperbolic in one dimension, and large degrees of sparsity did not notably affect accuracy on simple test cases.

    Source: §3.1 · p. 7, §3.2 · p. 10, §3.3 · p. 14

  • Takeaways & Limitations

    For the studied simple test cases, the method computed kinetic moment-equation solutions accurately while retaining well-posedness under kinetic boundary treatment and appropriate convection and timestepping schemes.

    Source: §6 · p. 30

    Strict hyperbolicity is established only in one dimension; in multiple dimensions, repeated eigenvalues occur generically, and the numerical tests use full three-dimensional distributions conserving all moments through total order Mmax.

    Source: §3.3 · p. 14, §5 · p. 22

Abstract

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In the present work, we study the properties of the sparse entropic quadrature method, a recently proposed method for the closure of systems of moment equations in rarefied gas dynamics. We prove the properties of the method regarding its well-posedness, hyperbolicity, and stability for first-order spatial convection schemes. We apply the method to two model rarefied gas dynamics problems and compare the results with those given by a discrete velocity method and show that it is capable of reproducing rarefied gas effects whilst having significantly fewer degrees of freedom than the discrete velocity method.

1 Introduction

Rarefied gas effects motivate moment models, but closing their equations requires estimating unknown higher-order moments while maintaining desirable mathematical and computational properties. The paper analyzes the sparse entropic quadrature method and applies it to one-dimensional flows.

1 Introduction

  • Modelling rarefied gas effects—those not captured by classical Euler and Navier–Stokes equations—is required to simulate a wide variety of problems in aerospace engineering, microelectronic-device design, and astrophysics.Source: §1 · p. 1
  • Moment closure requires estimating higher-order moments without knowing the distribution function, while ensuring non-negativity, hyperbolicity, conditioning, efficiency, and, in multiple dimensions, rotational invariance.Extending one-dimensional methods to multiple dimensions can impose additional constraints.Source: §1 · p. 2
  • Moment-closure approaches include Grad’s method, quadrature-based methods [22], maximum entropy methods, and unified formulations such as projection and φ-divergence.A separate study investigates convergence of moment equations to underlying kinetic equations [34].Source: §1 · p. 2
  • Hybrid methods combine ansatzes and optimization objectives; sparse entropic quadrature adds a sparsity-promoting term and uses local equilibrium as a preconditioner [41].Finite support avoids maximum-entropy non-realizability issues, while sparsity can support efficient storage and coupling with discrete-velocity solvers.Source: §1 · p. 2
  • The paper analyzes sparse entropic quadrature for moment closure and applies it to one-dimensional flows.It proves closure existence, uniqueness, and smoothness, analyzes one-dimensional strict hyperbolicity, and treats realizability for first-order transport schemes.Source: §1 · p. 3

2 Governing equations

The governing kinetic equation describes particle transport and collisions, while moment methods evolve integrals of the distribution to reduce the cost of solving it directly. Because moment fluxes require higher-order moments, the system needs a closure based on reconstructing a distribution from known moments.

2 Governing equations

  • The kinetic equation balances transport of the particle distribution with a collision operator; this work focuses mostly on its convective part.The distribution is a nonnegative probability density over position and velocity, and the collision operator describes particle interactions.Source: §2 · p. 3
  • Moment methods model the evolution of moments of the distribution because directly solving the high-dimensional kinetic equation is computationally expensive.The moments are transported through equations derived from the kinetic equation.Source: §2 · p. 3
  • A closure reconstructs a distribution from known lower-order moments to compute the higher-order moments needed for moment fluxes.The moment transport equations include flux and source terms.Source: §2 · pp. 3–4

3 Sparse entropic quadrature closure

This section develops the sparse entropic quadrature closure from moment constraints and establishes its solution, smoothness, and hyperbolicity properties. It examines characteristic speeds, wall mass-flux cancellation, and first- and second-order transport schemes designed to preserve realizability.

3 Sparse entropic quadrature closure

  • The closure writes f=w⊙g, using a local Maxwellian weight to improve conditioning and sparse g to reduce computation and storage and assess support reduction.The reconstruction is not exactly sparse because the stated formulation keeps f positive.Source: §3 · p. 5, §3.2 · p. 7

3.1 Existence and uniqueness of solution

  • Under assumptions 1–5 and for m in the realizability cone, the discrete primal problem has a unique positive solution, and strong duality holds with an attained dual optimum.The assumptions include an underdetermined, full-row-rank constraint matrix, a density constraint, positive weights, and a fixed velocity grid.Source: §3.1 · pp. 6–7, §3 · p. 6

3.2 Smoothness of the closure

  • For fixed w, the dual optimizer, reconstructed distribution, and higher-order moments are real-analytic in m; state-dependent Maxwellian weighting preserves analyticity but adds derivative corrections.When λ=0, the closure is independent of w and these correction terms vanish.Source: §3.2 · pp. 8–10

3.3 Hyperbolicity of the closed system in one dimension

  • For system (38)–(39), η is strictly convex on RM with ∇η = −y* and ∇²η = H⁻¹ ≻ 0, and (η, ψ) is an entropy/entropy-flux pair; F′(m) = KH⁻¹ makes the system symmetric hyperbolic.Source: §3.3 · p. 11
  • Frozen weights, including weights refitted once per time step and then held fixed, give a symmetric-hyperbolic system with distinct Gauss-node speeds strictly inside the velocity-grid bounds and a state-independent CFL bound.The characteristic speeds are the M-point Gauss quadrature nodes of the reconstructed measure.Source: §3.3 · pp. 12–13
  • Fully state-dependent weighting introduces an O(λ) symmetry defect, so strict hyperbolicity is conditional rather than unconditional.The perturbed system remains strictly hyperbolic when condition (52) holds.Source: §3.3 · pp. 13–14
  • In dimensions d ≥2, the system remains symmetric hyperbolic and retains the velocity-grid speed bound, but strictness generically fails because isotropy yields repeated Maxwellian-state eigenvalues.Source: §3.3 · p. 14

3.4 Numerical schemes for transport

  • The transport analysis considers one-dimensional x-direction transport on a uniform spatial grid while retaining a multidimensional distribution function.Source: §3.4 · p. 15

3.4.1 First-order schemes

  • The direct kinetic boundary treatment uses reconstructed distributions and an upwind boundary flux to make numerical wall mass flux vanish exactly.The resulting no-mass-crossing condition conserves mass globally for an internal flow.Source: §3.4.1 · pp. 17–18
  • Lax-Friedrichs, upwind, and global Rusanov first-order schemes preserve realizability when max_l |v_l,x| Δt/Δx ≤ 1.This CFL condition satisfies the admissibility requirement for the numerical viscosities.Source: §3.4.1 · p. 18
  • Under the CFL condition, first-order updates preserve a positive underlying distribution because each node update is a convex combination of positive reconstructed states.The reconstructed states remain positive for every λ ≥ 0 and positive w; boundary ghost states are also non-negative.Source: §3.4.1 · pp. 18–19
  • The same realizability argument applies to any update formed from node-wise convex combinations, including convex combinations of updates and SSP Runge–Kutta methods.The argument requires non-negative node-wise coefficients and the corresponding CFL condition.Source: §3.4.1 · p. 19

3.4.2 A flux-limited Lax–Wendroff scheme

  • The second-order scheme constructs a node-wise Lax–Wendroff flux for scalar advection and projects it to moments, yielding an exact kinetic-to-moment update.Its squared kinetic flux Jacobian shifts the monomial exponent by two in the x-direction, requiring one additional higher-order moment computation.Source: §3.4.2 · p. 20
  • For a positive limiter, the Lax–Wendroff update can lose positivity because one node-wise coefficient is negative when 0 < ν_l < 1.This prevents the unlimited kinetic update from being a convex combination of neighbouring states.Source: §3.4.2 · p. 21
  • A shared scalar van Leer limiter uses density and directional second-order moments, and can be reduced toward zero to keep updated moments realizable.The shared scalar confines the correction to the segment between the realizable first-order update and the unlimited second-order update.Source: §3.4.2 · p. 21

4 Overview of numerical algorithm for one-dimensional flows

The one-dimensional flow algorithm splits collision and convection, using a discretized Maxwellian that matches density, momentum, and energy. It integrates the BGK collision term exactly under a constant collision frequency and target distribution over each time step.

4 Overview of numerical algorithm for one-dimensional flows

  • The method uses a discretized Maxwellian matching density, momentum, and energy, found by Newton iteration; it is also the weighting distribution and the constrained-entropy target proposed in.Collisional effects use the BGK model.Source: §4 · p. 21
  • Operator splitting separates collision from convection, and the BGK collision term is integrated exactly assuming constant collision frequency and target distribution during each time step.The collision-driven moment change is expressed using ν and ∆t.Source: §4 · pp. 21–22
  • The collision frequency is computed using the variable hard sphere model, with an expression specified in scaled units.Its stated parameters include number density, the VHS exponent, reference viscosity, and reference temperature Tref = 1.Source: §4 · p. 22
  • Algorithm 1 outlines the first-order Lax–Friedrichs flow simulation; the upwinding scheme additionally precomputes A± and adjusts flux evaluation.The upwind moment measurement matrices are computed according to (65), with fluxes evaluated using (66) instead of (62).Source: §4 · p. 22

5 Numerical results

The Sod shock tube and Couette flow tests compare sparse entropic quadrature with a discrete velocity method across moment orders, numerical schemes, and regularization strengths. Higher moment orders improve agreement or boundary-layer resolution, while moderate sparsity can preserve solution quality; excessive regularization degrades solutions or prevents convergence.

5 Numerical results

  • The tests use a full three-dimensional velocity distribution and conserve all moments through Mmax = 2, 4, or 6, corresponding to 10, 35, or 84 moments.The study leaves conservation of different moment orders by dominant flow direction for future work.Source: §5 · p. 22

5.1 Sod shock tube

  • In the collisional Sod case, the 35- and 84-moment solutions agree better with DVM than the 10-moment solution, and 84 moments improve agreement in the low-density region.The 10-moment model lacks an explicit heat-flux term, which reduces heat transport and raises temperatures; Lax–Wendroff profiles are slightly sharper than upwind profiles.Source: §5.1 · p. 24, Figure 1 · p. 24
  • In the collisionless Sod case, all moment solutions deviate more from DVM than in the collisional case, although increasing moment order improves density, velocity, and temperature profiles.The discontinuous reference is poorly represented by the maximum-entropy closure; the 10-moment temperature profile is also poor because heat flux is not explicitly modeled.Source: §5.1 · pp. 24–25, Figure 2 · p. 25
  • For the collisional Sod case, λ = 10^-2 strongly degrades the Mmax = 4 solution and prevents convergence at Mmax = 6, while smaller λ values give better agreement with DVM.In the collisionless case, λ = 10^-2 prevents convergence for both moment orders, and lower regularization produces larger deviations than in the collisional case.Source: Figure 3 · p. 25, §5.1 · p. 26, Figure 4 · p. 26

5.2 Couette flow

  • For Couette-flow DVM profiles, upwind results are close across grids, whereas Lax–Friedrichs produces greater grid-dependent variation and qualitative shear-stress differences.The 100-cell upwind DVM result serves as the reference, with the 100-cell Lax–Friedrichs result retained for consistent flux comparisons.Source: §5.2 · p. 27, Figure 5 · p. 27
  • In Couette flow, Mmax = 2 deviates strongly from the reference, while Mmax = 4 and 6 give results close to DVM and Mmax = 6 improves boundary-layer resolution.Lax–Friedrichs also produces stronger shear-stress deviations; the flux-limited Lax–Wendroff solution exhibits slight center-domain oscillations.Source: §5.2 · pp. 28–29, Figure 6 · p. 28, Figure 7 · p. 29
  • About 80% of Couette weighting-function values fall below 10^-10 at λ = 10^-6 and 88–90% at λ = 10^-4, with results very similar to DVM for smaller λ; Sod remains accurate up to 90% sparsity.In both tests, higher sparsity degrades solution quality or destabilizes the solver; Couette sparsity could reduce degrees of freedom for coupling with DVM solvers.Source: §5.1 · pp. 26–27, §5.2 · p. 29

6 Conclusions

The sparse entropic quadrature method is well-posed, with its properties retained under kinetic boundary treatment and appropriate convection and time-stepping schemes. It accurately computes simple test cases, and substantial sparsity does not noticeably affect solution accuracy.

6 Conclusions

  • The method is well-posed, and this property is retained with kinetic boundary treatment coupled to appropriate convection and time-stepping schemes.Source: §6 · p. 30
  • The method accurately computes solutions of the kinetic moment equations for simple test cases.Enforcing even a large degree of sparsity does not noticeably affect solution accuracy.Source: §6 · p. 30
  • Future work will develop more sophisticated kinetic boundary treatments, more robust higher-order convection schemes, and applications to more complex problems.Source: §6 · p. 30
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