Unified Gas-kinetic Wave-Particle methods I: Continuum and Rarefied Gas Flow

1. Introduction

• Stochastic particle methods are efficient for rarefied and hypersonic flows but suffer statistical noise at low speeds and high costs near the continuum regime.Conventional splitting also restricts cell size and time step to below the mean free path and collision time, respectively.Source: §1 · p. 2
• Deterministic methods usually achieve high accuracy versus stochastic methods, but relatively long rarefied-flow simulations may suffer ray effects; hypersonic and multidimensional calculations are costly from direct velocity-space discretization, with cell-size and time-step constraints for accuracy.Source: §1 · pp. 2–3
• UGKS captures continuum Navier–Stokes solutions, including laminar boundary layers, with cells larger than the local mean free path and time steps larger than the local collision time.For time steps much larger than the collision time, its integral solution converges to a Chapman–Enskog-type Navier–Stokes distribution and yields the same solution as GKS.Source: §1 · pp. 4–5
• UGKS is a finite-volume scheme that updates the gas distribution function and macroscopic variables, with a CFL-limited time step and no mean-free-path or collision-time restriction.Its modeled dynamics depend on the time-step-to-collision-time and cell-size-to-mean-free-path ratios.Source: §1 · p. 4
• UGKS models time-dependent interface distributions and collision terms so transport and collision jointly shape interface evolution and provide a multiscale flux from initial to equilibrium states.In the continuum limit, this evolution converges to the Navier–Stokes distribution.Source: §1 · p. 5

Introduction

• The proposed wave-particle formulation aims to address particle methods’ statistical noise and discrete ordinate methods’ high computational cost, especially for hypersonic simulation.The paper applies the wave-particle formulation to solve the UGKS equations.Source: §1 · p. 6
• UGKWP uses probability density functions for hydro-particles and simulation particles for collisional and collisionless particles, coupling particle evolution with macroscopic-variable updates.Particle transport solves the integral equation within each cell, making a separate Eulerian update of the gas distribution function unnecessary.Source: §1 · p. 6
• UGKWP partitions gas particles into hydro-, collisional, and collisionless types whose proportions vary with flow regime and whose dynamics cycle between equilibrium and non-equilibrium descriptions.Hydro-particles are described analytically; collisionless particles mainly represent non-equilibrium transport, while hydro-particles represent equilibrium dynamics.Source: §1 · pp. 6–7
• In the continuum regime, collisional and collisionless particle counts decrease and UGKWP converges to GKS with the same computational cost.The introduction says its numerical flux converges to GKS flux in the hydrodynamic regime.Source: §1 · p. 7
• For hypersonic flows, UGKWP is described as more efficient than UGKS because simulation particles adapt naturally in particle velocity space.The introduction characterizes UGKWP’s computational cost as similar to particle methods in rarefied flow and reduced to that of a hydrodynamic solver in continuum flow, calling this multi-efficiency preservation.Source: §1 · p. 7
• Both UGKP and UGKWP are multiscale methods for all flow regimes, while UGKWP is more cost-efficient because hydro-particles have an analytical formulation.The paper introduces the multi-efficiency-preserving concept and analyzes UGKWP’s asymptotic properties.Source: §1 · p. 7

2. Unified Gas-kinetic Wave-Particle method

• UGKWP extends UGKP by representing collisional and collisionless particles explicitly while describing hydro-particles through an analytical distribution function.Source: §2 · p. 7

2.1. Unified gas-kinetic particle method

• UGKP tracks particles under the kinetic BGK equation until collision, then merges them into macroscopic quantities for resampling from updated flow variables.Source: §2.1 · p. 8

2.1.1. Evolution of particles

• Particles with first-collision times at or beyond the time step stream throughout it; earlier-colliding particles are tracked until collision, merged into cell quantities, and later resampled as hydro-particles.The first-collision time is sampled as tc = −τ ln(η), with η uniform on (0, 1).Source: §2.1.1 · p. 9

2.1.2. Evolution of macroscopic flow variables

• Macroscopic variables evolve through UGKS fluxes, with Maxwellian-related interface terms and free-transport terms calculated from simulation particles.Source: §2.1.2 · pp. 11, 13
• The end-of-step count includes collisionless particles only, whereas the next step’s total also includes newly sampled hydro-particles.Source: §2.1.2 · p. 13

2.2. Unified gas-kinetic wave-particle method

• UGKWP analytically evaluates hydro-particle free-transport flux, eliminating the need to sample all hydro-particles.The method uses the known hydro-quantity distribution to evaluate this flux.Source: §2.2 · p. 14
• UGKWP samples only collisionless hydro-particles, using the first-collision-time distribution to draw them from a Maxwellian.The method distinguishes collisionless particles, collisional particles, and collisionless hydro-particles.Source: §2.2 · p. 14
• Collisionless particles and collisionless hydro-particles advance, collisional particles are removed after flux calculation, and the remaining particles are reassigned by sampled collision time.Figure 2 depicts the dynamic circulation among particle-described collisionless and collisional particles, wave-described free-transport hydro-particles, and macroscopic variables.Source: §2.2 · p. 15, Figure 2 · p. 15

3.1. Asymptotic behavior in continuum regime

• For Δt≫τ, UGKWP preserves hydrodynamic Navier–Stokes solutions with first-order Chapman–Enskog-consistent flux, while particle count falls as O(e^-Δt/τ) and cost becomes comparable to NS solvers.The continuum analysis identifies the particle-count decay and solver-cost comparison in this limit.Source: §3.1 · p. 17
• MMP requires preserving collisionless, Navier–Stokes, and Euler solutions without sub-mean-free-path cell-size or time-step restrictions, alongside efficiency comparable to high-efficiency schemes.The paper identifies UGKWP as an MMP scheme.Source: §3.1 · pp. 17–18

3.2. Consistent sampling

• Hydro-particles are sampled from a Maxwellian, then velocity-transformed so their macroscopic quantities match the specified sampling quantities.The transformation parameters are solved from consistent constraints.Source: §3.2 · p. 18

3.3. Robustness and conservation

• Statistical noise can make cell-averaged density and temperature negative, so corrections are applied in both macroscopic-variable updates and hydro-quantity calculations.The method is a finite-volume scheme for macroscopic variables.Source: §3.3 · p. 19
• Mass, momentum, and energy are conserved when Eq.(44) is inactive and the corrected density and energy equal their cell values; Eq.(44) was inactive in all reported tests.The authors describe it as a safety safeguard for when needed.Source: §3.3 · p. 19

4. Numerical tests

• The five tests cover Sod shock tube, normal shock, cylinder flow, lid-driven cavity, and flat-plate boundary layer; results show agreement with references across speeds and regimes.The section overview also reports higher hypersonic-flow efficiency and lower memory cost than direct-velocity-discretization UGKS, and lower near-continuum statistical noise than conventional particle methods.Source: §4 · p. 19

4.1. Sod shock tube problem

• At Kn = 0.1, UGKWP with 10^4 particles per cell agrees with UGKS, while reducing the count to 10 per cell produces extreme statistical noise; sufficient particles or averaging are required.Source: §4.1 · p. 20

4.2. Normal shock

• For the normalized solution, UGKWP agrees well with the UGKS solution, supporting the method’s demonstrated ability to capture non-equilibrium flow.Source: §4.2 · p. 21

4.3. Cylinder flow

• UGKWP required 21.5 minutes, including the averaging procedure, versus 10.9 hours for UGKS, and its memory cost was 75 MB versus UGKS’s 3.4 GB.The reported comparison described UGKWP as about 30 times faster than explicit UGKS.Source: §4.3 · p. 22
• UGKS took about 429 hours, while UGKWP took 36.1 minutes (including the averaging procedure), making UGKWP about 713 times faster than explicit UGKS.UGKS memory cost was 22.3 GB, compared with less than 100 MB for UGKWP; UGKWP’s memory cost was 228 times less.Source: §4.3 · p. 22

4.4. Cavity flow

• The lid-driven cavity tests assess multiscale low-speed flow at Kn = 1.0, 0.75, and 1.42 × 10^-4.The setup assumes monatomic argon modeled with the VHS model, wall temperature Tw = 273 K, and lid velocity Uw = 50 m/s.Source: §4.4 · p. 23
• For Kn = 1.0, UGKWP took 20 hours and 2.5 GB, making it 1.38 times slower and 5 times more memory-intensive than UGKS.UGKWP’s time includes 1800 averaging steps; UGKS took 14.4 hours and used 500 MB.Source: §4.4 · p. 23
• For Kn = 1.0 and Kn = 0.075, UGKWP density and velocity contours agree well with UGKS, but temperature contours show relatively large statistical noise.The velocity profiles also show good agreement between the methods.Source: §4.4 · pp. 23–24
• UGKWP takes 38.7 minutes versus 11 hours for explicit UGKS, is 17 times faster, and uses 26 times less memory; its velocity profiles agree well with the Navier–Stokes solution.The memory cost is 13 GB for UGKS and 500 MB for UGKWP.Source: §4.4 · p. 24

4.5. Boundary layer

• The UGKWP velocity profile agrees well with the Blasius solution; with a time step much larger than local collision time, each cell stores about 1–2 particles, and computation takes less than 2 minutes.The test addresses calculating a Navier–Stokes solution when cell size is much larger than mean free path and time step is much larger than collision time.Source: §4.5 · p. 24

• The method couples a probability density distribution with simulation particles, sampling particles to capture local non-equilibrium from free transport while evolving microscopic and macroscopic quantities together.Source: §5 · p. 25
• UGKWP captures flow dynamics efficiently across regimes and is described as applicable from hypersonic flows to low-speed microflows.Source: §5 · p. 25
• UGKWP is multi-efficiency preserving: its cost is on the scale of particle methods in rarefied flow and comparable to hydrodynamic solvers in continuum flow.Source: §5 · p. 25
• For hypersonic flows, UGKWP reduces computational cost by hundreds of times versus UGKS with direct discretization of particle velocity space.Source: §5 · p. 25
• The current method uses a BGK-type collision model, while more realistic collisional models can be included in UGKWP.Source: §5 · p. 25
• Particle streaming or fluid-element penetration can capture non-equilibrium effects, and the direct-modeling equations release the continuum-mechanics constraint and may help study non-equilibrium turbulent flow.Source: §5 · pp. 25–26