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A Gradient-Reconstruction Lattice Boltzmann Method for Compressible Navier--Stokes--Fourier Equations

Adrian Kummerländer, Fedor Bukreev, Mathias J. Krause

arXiv:2609.16361v1physics.comp-phcs.MS

TL;DR

Compressible lattice Boltzmann methods have typically given up a compact stencil or strict locality. This method reconstructs closure gradients from non-equilibrium moments during collision, and matches exact shock solutions while staying within the turbulence reference band on key measures.

  • Problem

    Compressible lattice Boltzmann methods have typically given up a compact stencil or strict locality to reach compressible flow.

    Source: §1 · p. 1

  • Method

    The scheme reconstructs velocity and temperature gradients from conserved fields’ non-equilibrium moments during collision, transporting only five conserved fields on D3Q7.

    Source: §1 · pp. 1–2

  • Results

    The five-field scheme matches the fourteen-field baseline on exact Sod and Becker solutions and stays within the seven-solver reference band on kinetic energy and both dissipation rates for the compressible Taylor–Green vortex.

    Source: §5 · p. 12

  • Takeaways & Limitations

    The reconstruction is the efficient choice where memory and time bind, while the relaxed-flux scheme remains more accurate per grid point in dissipation channels.

    Source: §5 · p. 12

    Some reference data were digitized from raster figures.

    Source: §3.3 · p. 8

Abstract

from arXiv · show

Reaching compressible flow has usually forced lattice Boltzmann methods to abandon the compact stencil or the strict locality that make them efficient. We give up neither, solving the compressible Navier--Stokes--Fourier equations with a scheme that transports only the conserved mass, momentum and energy. The viscous stress and heat flux depend on gradients of the velocity and temperature. To capture them, existing compressible schemes go beyond a single lattice of the conserved fields. They enlarge the velocity set, carry the stress and heat flux as extra transported fields, place the energy on a separate grid, or give up exact streaming for an off-lattice advection. We instead recover the gradients from the non-equilibrium part of the distributions already present, inside the collision. No field beyond the conserved state is transported, no neighbour is read, streaming stays exact, and every computation runs in single precision. The method carries five fields where a transported-flux scheme carries fourteen, at a fraction of the memory and $4.9$ times the throughput. Verified against the exact Sod and Becker shock solutions and a supersonic Taylor--Green vortex, it tracks the direct-numerical-simulation reference on the kinetic energy and stays within the reference-solver spread on both dissipation rates.

1. Introduction

The method reconstructs velocity and temperature gradients during collision, allowing compressible Navier–Stokes–Fourier flow on a compact, exactly streaming lattice while transporting only conserved fields. It is validated against exact shock solutions and a supersonic turbulence reference, with reported accuracy and throughput advantages.

1. Introduction

  • Earlier compressible lattice Boltzmann approaches enlarge the velocity stencil, give up exact on-lattice streaming, use hybrid solvers, or transport an extended state.For the extended-state route, the stress and heat flux are auxiliary fields supplying gradients required by the physical closures.Source: §1 · p. 1
  • The method recovers closure gradients from conserved fields’ non-equilibrium moments during collision, using a diffusive-flux Jacobian operator on D3Q7.It reaches supersonic shocklet-laden turbulence with exact streaming and no auxiliary state.Source: §1 · pp. 1–2
  • The reconstruction reduces the transported state from fourteen fields and ninety-eight streamed populations to five fields and thirty-five populations.Stress and heat flux are formed during collision rather than transported as relaxed states.Source: §1 · p. 2
  • Validation covers exact shock solutions and a supersonic DNS turbulence reference, with results staying within the reference band.The benchmarks include Sod, Becker’s traveling wave, and the compressible Taylor–Green vortex.Source: §1 · p. 2
  • The method is the most accurate second-order scheme in the comparison, ahead of the only other one in every channel, and runs entirely in single precision.Source: §1 · p. 2
  • 4.9 times the throughput of the fourteen-field relaxed-flux scheme, with the five-field method reaching 5123 on a two-GPU node.It transports only the conserved fields.Source: §1 · p. 2

2. Methodology

The method transports the five conserved fields, reconstructs closure gradients during collision, and uses a local compression-triggered regularized collision for shock cases.

2.1. Governing Equations

  • The compressible system transports only the five conserved fields; Newtonian stress and Fourier heat flux are evaluated from reconstructed gradients rather than carried as additional states.The relaxed-flux scheme carries stress and heat flux as nine additional states.Source: §2.1 · p. 2

2.2. Discretization

  • Each conserved field has its own populations, with the flux in the equilibrium’s first moment and the state in its zeroth; D3Q7 gives 35 populations per cell.Smooth cases use BGK, while shock tubes and the supersonic vortex use the RLB operator.Source: §2.2 · p. 2

2.3. Gradient Reconstruction

  • The non-equilibrium first moment carries each conserved gradient to leading order; subtracting advection and applying the closed-form Jacobian inverse reconstructs gradients in-collision, then the chain rule yields ∇u and ∇T.The reconstruction is explicit within the step; the stable timestep has a Courant ceiling of order 0.7.Source: §2.3 · pp. 2–3

2.4. The Compressible Reconstruction Operator

  • The state-dependent, block-lower-triangular inverse passes ∇ρ unchanged, jointly corrects symmetric-deviatoric momentum gradients, then solves energy with viscous-work and Fourier-conduction coupling.At Taylor–Green conditions β≈0.02 and R is within a few percent of identity; at manufactured-solution and shock-tube points β grows from 0.5 at N=64 to 4 at N=512.Source: §2.4 · pp. 3–4

2.5. Shock Capturing

  • A cell-local compression sensor gates extra damping, and the RLB collision regularizes the non-equilibrium while lowering its relaxation rate where the sensor fires.The sensor uses reconstructed dilatation, activates only in compression above a threshold, and leaves smooth regions at the physical rate.Source: §2.5 · p. 4

2.6. Numerical Parameters

  • ω and CFL tune stability and accuracy without changing physical transport; accuracy favors ω→2, while stability is bounded near CFL≈0.7, from about 0.8 at Res=250 to 0.65 at Res=4000.The numerical parameters are collected for every validation case in Table 1.Source: §2.6 · pp. 4–5, Table 1 · p. 5

3. Validation

Validation spans manufactured solutions, exact Sod and Becker shocks, and a supersonic Taylor–Green vortex; results establish second-order smooth-field convergence and reference-band turbulence behavior.

3.1. Manufactured Solution

  • Manufactured solutions show second-order convergence for all five conserved fields, with single precision indistinguishable from double precision to five significant figures.The test uses a smooth, unsteady supersonic field at M0 = 2.Source: §3.1 · pp. 5–6, Table 2 · p. 6

3.2. Shock Tube

  • Sod profiles remain monotone without overshoot, while Becker density error falls from 1.3e-1 at 8 front cells to 1.6e-3 at 128, reaching order 1.9 by 64.The Sod test is compared with the exact Riemann solution; Becker is compared with the exact traveling wave.Source: Table 4 · p. 6, §3.2 · p. 6, Table 3 · p. 6

3.3. Compressible Taylor–Green Vortex

  • At both resolutions, the five-field scheme stays within the reference band on all three channels; at 512^3, relaxed-flux has lower errors on all three.The channels are kinetic energy and solenoidal and dilatational dissipation rates.Source: §3.3 · p. 7, Table 5 · p. 8
  • The five-field spectrum matches the fourteen-field scheme at large scales and rolls off slightly faster at high k; refinement extends the resolved range without grid-scale pile-up.The refinement comparison is from 256^3 to 512^3.Source: §3.3 · p. 8

4. Performance

The five-field scheme delivers higher throughput with a smaller memory footprint than the fourteen-field relaxed-flux scheme, while trading a modest dissipation-accuracy margin. The performance evaluation also compares turbulence, dissipation, flow-profile, and spectral behavior across the supplied figures.

Memory traffic

  • The five-field scheme moves 284 B per cell versus 784 B for relaxed-flux, a 2.8× reduction, with reconstructed gradients formed in registers and adding no traffic.Both schemes use D3Q7; the five-field total includes the per-cell shock-sensor read.Source: §4 · p. 8

Time to solution and problem size

  • At 512^3, the five-field state is 19 GB and fits the two-GPU node, whereas the 14-field state is 53 GB and does not; relaxed-flux is more accurate per grid point in dissipation channels.At 256^3, the schemes have comparable accuracy, so their wall-clock ratio is the 4.9× throughput ratio.Source: §4 · p. 8

Benchmark figures

  • For Re = 1600, M0 = 1.25, figures compare kinetic energy and dilatational and solenoidal dissipation over time across four resolutions and reference schemes; they also show Mach profiles and velocity spectra.The velocity spectra are shell-averaged; the 512^3 solenoidal-dissipation curves for three damping values separate only in the late tail.Source: Figure 5 · p. 9, Figure 6 · p. 10, Figure 7 · p. 10

Throughput

  • 4.9× speed-up: at 256^3, the five-field scheme sustains 2028 MLUP/s versus 413 MLUP/s for relaxed-flux on the same GPU in single precision.The five-field kernel reaches 85% of peak bandwidth, compared with 48% for relaxed-flux.Source: §4 · p. 8, Table 6 · p. 9, Figure 11 · p. 12

Roofline

  • The roofline places both kernels in the memory-bound region, and reducing the state from fourteen fields to five moves the kernel toward the roof.Figure 11 marks the ridge at arithmetic intensity I = 40.9.Source: Figure 11 · p. 12

5. Conclusion

The conclusion presents a five-field CNSF lattice Boltzmann method that reconstructs gradients during collision and validates it against shock solutions and compressible turbulence references. It reports substantially lower memory use and higher throughput, with relaxed-flux retaining an accuracy advantage per grid point in dissipation channels.

Method

  • The method transports only the conserved state, recovering velocity and temperature gradients from non-equilibrium first moments during collision to form viscous stress and heat flux.It uses five fields on D3Q7 with exact streaming in single precision, compared with fourteen fields for the relaxed-flux scheme.Source: §5 · p. 12

Validation

  • The five-field scheme matches the fourteen-field baseline on exact Sod and Becker solutions and stays within the seven-solver reference band on kinetic energy and both dissipation rates.Against H3AMR, the only other second-order scheme in the comparison, it is more accurate in every channel at both 256^3 and 512^3.Source: §5 · p. 12

Performance

  • A 2.8× traffic reduction accompanies a 4.9× speed-up: the five-field scheme moves 284 B per cell versus 784 B and sustains 2028 versus 413 MLUP/s.At 512^3, the five-field state is 19 GB and fits the tested node, while the 53 GB fourteen-field state does not; relaxed-flux remains more accurate per grid point in dissipation channels.Source: §5 · p. 12

Contribution

  • The CNSF system joins those solvable by reconstruction from a single declaration rather than by transporting auxiliary state.The cited reconstruction work is.Source: §5 · p. 12

Code Availability

The paper specifies the five-field scheme and identifies how to obtain the compiler and simulation cases.

Availability

  • The five-field scheme is fully specified through the declaration in Listing 1, the reconstruction operator in Section 2.4, the shock-capturing collision in Listing 2, and the benchmark configurations in Section 3.The generated collide-and-stream kernels run in OpenLB release 1.9.Source: Code Availability · p. 12

Framework

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