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Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations?
Wei Su, Lianhua Zhu, Peng Wang, Yonghao Zhang, Lei Wu
TL;DR
Steady-state Boltzmann solutions are especially slow to obtain in near-continuum flows, despite the equation’s applicability across Knudsen regimes. This paper proposes GSIS, combining kinetic and synthetic macroscopic equations, and reports convergence in about 20 iterations across Knudsen numbers.
Problem
Steady-state Boltzmann solutions remain slow to obtain in near-continuum flows, including with UGKS and hybrid approaches that still use kinetic methods in low-Knudsen regions.
Source: §1 · p. 3Method
GSIS solves the gas kinetic equation together with synthetic macroscopic equations for general rarefied flows.
Source: §1 · p. 4, §6 · p. 33Results
About 20 iterations suffice to obtain converged linearized Boltzmann solutions at any Knudsen number, according to the reported numerical results.
Source: §6 · p. 33Takeaways & Limitations
The authors state that GSIS does not depend on a specific collision operator and can be extended to nonlinear Boltzmann equations.
Source: §6 · p. 33The segregated SIMPLE approach used to solve the synthetic equations can require several hundred iterations, and its per-iteration cost is non-negligible.
Source: §4.2 · p. 18
Abstract
from arXiv · showhide
One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iteration scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the near-continuum regime, hundreds of thousands iterations are needed, and yet the "converged" solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implicit unified gas kinetic scheme (UGKS) and its variants have significantly reduced the iterations in the near-continuum flow regime, but still much higher than that of the highly rarefied gas flows. In this paper, we put forward a general synthetic iteration scheme (GSIS) to find the steady-state solutions of general rarefied gas flows within dozens of iterations at any Knudsen number. As the GSIS does not rely on the specific kinetic model/collision operator, it can be naturally extended to quickly find converged solutions for mixture flows and even flows involving chemical reactions. These two superior advantages are also expected to accelerate the slow convergence in simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version.
1. Introduction
The Boltzmann equation describes gas flows across the full range of Knudsen numbers, but finding steady states becomes costly and slow near the continuum regime. GSIS builds on synthetic iterative schemes to target general rarefied flows, extending beyond the simpler cases previously studied.
1. Introduction
- The Boltzmann equation describes the molecular velocity distribution and applies across the entire range of Knudsen numbers.By contrast, the Navier–Stokes model is valid only when macroscopic gradients vary over scales much larger than the molecular mean free path.Source: §1 · p. 2
- Boltzmann solvers such as DVM and DSMC are more costly than conventional CFD because they require molecular velocity discretization or particles, along with fine spatial and temporal resolution.These resolution requirements become especially costly as flows approach the near-continuum regime.Source: §1 · p. 2
- Near-continuum Boltzmann calculations are costly: conventional iteration converges slowly, errors can accumulate, and UGKS still requires many steady-state iterations.Hybrid continuum-kinetic methods also leave the Boltzmann equation in low-Knudsen regions, where convergence and DSMC evolution remain expensive.Source: §1 · p. 3
- Prior synthetic iterative schemes can converge quickly and preserve high-Knudsen accuracy, but had been applied to simpler rarefied flows rather than general flow geometries.The cited examples include Poiseuille, Couette, and thermal transpiration flows with velocity perpendicular to the computational domain.Source: §1 · pp. 3–4
- GSIS aims to find steady-state solutions for general rarefied flows within dozens of iterations at any Knudsen number, with the present study focusing on linearized flows.The authors state that the methodology can be extended to nonlinear flows.Source: §1 · p. 4
2. The linearized Boltzmann equation
The section formulates a linearized Boltzmann equation for a perturbation around equilibrium and specifies collision and boundary modeling used in the paper. The perturbation is small relative to equilibrium, although the function h itself need not be.
2. The linearized Boltzmann equation
- The distribution is linearized around equilibrium using a small perturbation, but h itself need not be smaller than the equilibrium distribution.The perturbation satisfies |αh/feq| ≪ 1.Source: §2 · p. 4
- The collision kernel is modeled for inverse power-law potentials, with its cross-section determined by the intermolecular potential.The paper notes that the fast spectral implementation for Lennard-Jones potentials is not used here; only the collision kernel is used to demonstrate GSIS for the linearized Boltzmann equation.Source: §2 · p. 5
- The governing variables are dimensionless, with position, velocity, time, and distribution functions normalized by the stated reference scales.The paper specifies H, the most probable speed, H/vm, and n0/v^3m as the respective scales.Source: §2 · p. 5
- For gas-surface interactions, the paper uses the Maxwell diffuse boundary condition for reflected molecules.The wall temperature and velocity are normalized by the reference temperature and most probable speed.Source: §2 · p. 5
3. The general synthetic iteration scheme
GSIS couples kinetic iteration with synthetic macroscopic equations whose stress and heat-flux relations explicitly recover the Navier–Stokes transport laws while retaining higher-order rarefaction terms. This coupling is intended to speed convergence at low Knudsen numbers without sacrificing the kinetic description of general rarefied flows.
3. The general synthetic iteration scheme
- The conventional iteration scheme becomes inefficient as the rarefaction parameter increases because frequent molecular collisions hinder information exchange across the flow.Near-continuum flows need macroscopic synthetic equations that preserve the Navier–Stokes limit to improve that exchange.Source: §3 · p. 6
- Higher-order macroscopic equations are unstable or, even when stable, approximate rather than exact solutions of the Boltzmann equation.This motivates using higher-order terms computed from the distribution function rather than relying on a truncated macroscopic closure.Source: §3 · pp. 7–8
- GSIS explicitly incorporates Newton’s stress law and Fourier’s heat-conduction law into its synthetic equations, with higher-order terms accounting for rarefaction effects.The scheme derives the synthetic stress and heat-flux equations from moments of the linearized Boltzmann equation.Source: §3 · pp. 7–8
- The synthetic macroscopic equations preserve the Navier–Stokes limit while retaining higher-order terms computed from the velocity distribution, avoiding the approximations of Grad 13 closure.Their Chapman–Enskog first-order limit recovers Navier–Stokes behavior, while higher-order contributions account for rarefaction effects.Source: §3 · pp. 8–9
- Each GSIS iteration first advances the distribution conventionally, computes macroscopic quantities and higher-order terms, solves the synthetic equations, then adjusts the distribution to match the updated quantities.The process repeats until convergence; conventional spatial discretizations and fast spectral collision calculations may be used.Source: §3 · pp. 9–10
- GSIS solves the kinetic equation together with synthetic macroscopic equations for general rarefied flows, and the authors state the approach can extend directly to the nonlinear Boltzmann equation.They describe this as the first GSIS developed for general rarefied flows described by the linearized Boltzmann equation.Source: §3 · p. 10
4. Numerical results for stationary problems
The GSIS is tested on Fourier heat transfer, a lid-driven cavity, and shear-driven flow between eccentric cylinders across rarefaction regimes. Across these cases, it combines rapid convergence with accuracy on coarse grids.
4.1. Heat transfer between two parallel plates
- For the Shakhov model at δrp = 50, GSIS made bulk density and temperature profiles nearly linear after the first iteration, enabling faster convergence across the domain.CIS took about 50 iterations for the bulk temperature to register the wall perturbation, and its error oscillated before decaying.Source: §4.1 · pp. 12–13, Figure 1 · p. 12
- Across rarefaction parameters, GSIS converged in 20 iterations for Fourier flow, while CIS iteration counts rose nearly as δrp^2 and its error oscillated at higher δrp.At δrp = 50, GSIS was about 100 times more efficient than CIS; its error decreased monotonically across the tested regimes.Source: §4.1 · p. 13, Figure 2 · p. 13
- In the CIS, profile and heat-flux relative errors rise from 0.3% to 9% and 16%, respectively; GSIS relative error remains within 1%, even when cells are about five times the gas mean free path.At δrp = 500, GSIS heat flux changes from 3.721 × 10−3 at N2 = 551 to 3.726 × 10−3 at N2 = 6; the authors attribute its performance to asymptotically preserving the Navier-Stokes limit.Source: §4.1 · p. 14
- For the linearized Boltzmann equation, GSIS reached steady state in 22 iterations with hard-sphere molecules and 27 with Maxwell molecules; temperature-jump coefficients were 1.892 and 1.954.The Knudsen layer functions differed more between collision models, and the authors note that a small term in the model significantly affects them.Source: §4.1 · p. 15, Figure 4 · p. 15
4.2. Two-dimensional lid-driven cavity flow
- For lid-driven cavity flow, GSIS required fewer than 40 iterations at δrp = 1000, while CIS required 1,823 at δrp = 100 and was not run at 1000.On the coarsest grid, GSIS was more accurate than CIS; GSIS captured short contours near the bottom wall that CIS captured only on its finest grid.Source: §4.2 · pp. 17–19, Figure 6 · p. 18
4.3. Shear-driven flow between two eccentric cylinders
- 26 GSIS iterations versus 49,454 CIS iterations at δrp = 1000, and 26 versus 296 at δrp = 10, were reported for eccentric-cylinder flow.The GSIS was nearly 2000 and 10 times faster, respectively; at δrp = 1000 it also matched the Navier-Stokes result on a mesh with cells up to 260 mean free paths high, unlike CIS.Source: Figure 8 · p. 20, §4.3 · p. 21
5. Numerical results for periodic oscillation problems
Synthetic macroscopic equations extend GSIS to periodic oscillation problems, with examples spanning scattering, plate flows, and a cavity. Across these cases, the reported results show fast convergence and accurate solutions, including where CIS is sensitive to flow parameters or spatial resolution.
5. Numerical results for periodic oscillation problems
- Periodic forcing is handled by replacing ∂/∂t with iSt in the synthetic macroscopic equations.The paper demonstrates this approach on SRBS, oscillatory Couette flow, and sound propagation.Source: §5 · p. 21
5.1. Spontaneous Rayleigh-Brillouin scattering
- At δrp=500, GSIS converges within 20 steps at every frequency; at δrp=10 and 50 it uses about 10 and 100 times fewer iterations than CIS.At δrp=50, CIS also produces a false-converged discontinuous spectrum near frequency difference 0.68, while the GSIS and CIS spectra otherwise agree closely.Source: §5.1 · p. 23
- At small δrp and large St, direct GSIS may fail or blow up; replacing δrp on the synthetic equation's left-hand side with max(δrp, 10) damps nonphysical oscillations without changing the converged solution.The modified system converges to a solution satisfying the original equation.Source: §5.1 · p. 23
5.2. Oscillatory Couette flow between two parallel plates
- At δrp=50, GSIS needs about 20 iterations across tested Strouhal numbers, with shear-stress amplitudes within 1% of CIS.CIS iterations fall from 30,000 to 100 as St rises from 0 to 50; GSIS remains near 20 iterations. GSIS also gives accurate results with cell sizes about 6.6 and 50 times the mean free path, where CIS has large numerical-dissipation error.Source: §5.2 · pp. 26–27, Figure 12 · p. 26
5.3. Sound propagation between two parallel plates
- For sound propagation, CIS iteration counts scale roughly as St^-1.5 at small St, whereas GSIS iteration counts have little dependence on St and require fewer spatial grids.At St=2.5, the GSIS profiles agree with CIS results across the tested spatial discretizations.Source: Figure 14 · p. 29, §5.3 · p. 29, Figure 15 · p. 30
5.4. Two-dimensional oscillatory Couette flow
- For δrp=50, cavity shear stress nearly saturates across a broad St range at small aspect ratios; at large St, average shear force differs from the one-dimensional case by less than 5%.At Asp=2, average shear force is the same as for Asp=∞ except that it is slightly larger at small St; smaller aspect ratios widen the near-constant range.Source: §5.4 · p. 31, Figure 16 · p. 32
6. Conclusions and outlooks
The paper concludes that GSIS efficiently and accurately finds steady-state linearized Boltzmann solutions across Knudsen numbers. Its synthetic macroscopic equations support fast convergence and accurate results on cells larger than the mean free path, and the authors outline extensions to DSMC and more complex flows.
6. Conclusions and outlooks
- GSIS finds converged linearized Boltzmann solutions in about 20 iterations across Knudsen numbers, with accurate results on cells much larger than the mean free path and without large numerical dissipation.The authors attribute fast convergence to synthetic equations that preserve the Navier–Stokes limit and explicitly include first-order shear-stress and heat-flux constitutive laws.Source: §6 · p. 33
- Unlike implicit UGKS variants, GSIS does not depend on a specific collision operator and recovers Navier–Stokes explicitly, avoiding complex flux evaluation.The paper also reports that the framework has been implemented for nonlinear Fourier heat transfer, with convergence within 20 iterations at arbitrary Knudsen number.Source: §6 · p. 33
- The authors propose applying GSIS to low-variance and frequency-domain DSMC, especially near the continuum regime, and extending it to multi-species and compressible flows.They identify constructing macroscopic equations that recover compressible Navier–Stokes to first order in Knudsen number as the key to those extensions.Source: §6 · p. 33
- The authors also suggest applying the synthetic-iteration idea to the Enskog equation for dense-gas dynamics, including shale-gas extraction and non-equilibrium evaporation and condensation.These are presented as possible future applications.Source: §6 · p. 34
Appendix
The appendix describes an HDG discretization of the steady-state governing equations. It solves globally for traces of the flow properties, then recovers element-level fields and auxiliary variables locally.
Appendix
- Auxiliary variables approximate velocity and temperature gradients together with high-order moments, supporting evaluation of the stress tensor and heat flux.The stress tensor and heat flux are calculated after the local problem is reconstructed.Source: Appendix · pp. 34, 36
- The HDG procedure first determines the traces of flow properties globally, then solves element-by-element for the fields and auxiliary variables using those traces as boundary conditions.The traces are single-valued on each face.Source: Appendix · p. 35
- Eliminating element-level degrees of freedom reduces the assembled system to one involving only the trace unknowns; the local fields are reconstructed after that system is solved.The element-level coefficient blocks have block-diagonal structure, enabling the elimination.Source: Appendix · p. 36