The unified gas-kinetic wave-particle (UGKWP) method is a hybrid method for multiscale flow simulations, in which the contributions to the whole gas evolution from deterministic hydrodynamic wave and stochastic particle transport are combined simultaneously. Originally, the UGKWP method was developed as a direct modeling approach at discrete level. In this work, we revisit the time evolution of each part of the involved simulation particles and wave molecules in UGKWP, and present the corresponding kinetic equations. The resultant kinetic system can be viewed as a collision decomposition of the original kinetic equation, which can serve as a basis for developing other kinetic methods for flows in all flow regimes.

This study presents a systematic performance evaluation of three flux-reconstruction schemes in the discrete unified gas-kinetic scheme (DUGKS) for low-speed continuum flows. The collision term integral at cell interfaces is approximated with the left-rectangular, trapezoidal (standard DUGKS) and right-rectangular rules, while a unified finite-volume framework is retained elsewhere. Theoretical analysis confirms that all schemes retain second-order accuracy in time and space; however, numerical benchmarks (Taylor-Green vortex flow, lid-driven cavity flow, and flow past a square cylinder) reveal clear performance differences. On the same meshes, the right-rectangular and trapezoidal rules produce nearly equal errors, with the right-rectangular rule marginally superior, whereas the left-rectangular rule is an order of magnitude less accurate. Per time step, the rectangular rules are faster than the trapezoidal rule; when total runtime to convergence is considered, the right-rectangular rule is the most economical and markedly more stable, remaining robust for exceptionally large ratios between the time step and the relaxation time. Hence, the right-rectangular formulation offers the best overall balance of accuracy, efficiency, and stability for continuum flow simulations.

Accurate modelling of confined fluid transport requires addressing the fluid-fluid and fluid-surface interactions across multiple scales, particularly when molecular sizes are comparable to the fluid mean free path and the confinement dimension. Advances in nanotechnology have sparked tremendous interest in the combined effects of non-equilibrium, real-fluid properties, and confinement on multiscale flows, where the Knudsen number or confinement dimension can vary widely. This review discusses molecular kinetic modelling approaches for: (1) fluid density ranging from dilute, where non-equilibrium effects dominate, to dense, where real fluid effects become critical; (2) flow domains ranging from macroscale to nanoscale, necessitating confinement-specific treatments; (3) fluid-surface interactions at various densities and confinements. The pronounced non-equilibrium and confinement effects introduce both intrinsic and apparent non-hydrodynamic effects, which require careful consideration in developing a molecular kinetic model.

We develop an efficient discrete unified gas kinetic scheme (DUGKS) to solve a kinetic model for strongly inhomogeneous fluid flows at the nanoscale. The proposed DUGKS adopts efficient numerical strategies to evaluate the multiple nonlocal integrals involved in the kinetic equation, reducing the computational cost from O(NN sigma) for the direct evaluation to O(N), where N and N sigma are the number of cells in the flow domain and integral domain, respectively. The accuracy and efficiency of the proposed DUGKS are assessed through several test cases, including static fluid structures and force-driven flow in nano slits. The present results agree well with those from Monte Carlo and molecular dynamics simulations, as well as with those from the original DUGKS with direct evaluation of the integrals. Moreover, the speedup relative to the original DUGKS can reach up to two orders of magnitude. As example applications, we investigate two-dimensional pressure-driven flow between parallel plates and force-driven flow in a square duct, revealing distinctive nanoscale features such as the non-equivalence of pressure-and force-driven flows and the formation of density peaks in the square duct corners.

Inferring thermal fields and thermophysical properties from limited measurements is a fundamental challenge in micro- and nanoscale heat conduction, where the classical Fourier law breaks down and the phonon Boltzmann transport equation (BTE) is needed to capture non-diffusive transport effects. In this work, we extend Monte Carlo physics-informed neural networks (MC-PINNs), originally developed for forward phonon BTE problems [J. Comput. Phys. 542, 114364, 2025], to inverse multiscale heat conduction problems. Two representative classes of inverse problems are considered: (i) reconstructing the full thermal field from sparse interior temperature measurements when boundary conditions are unknown, and (ii) simultaneously inferring the unknown relaxation time together with the thermal field. Problem-specific MC-PINN architectures and training strategies are designed for each class. The mesh-free Monte Carlo sampling strategy enables a unified treatment across diffusive, transitional, and ballistic transport regimes without requiring a priori knowledge of the relaxation time. The proposed method is evaluated on quasi-one-dimensional, quasi-two-dimensional, and three-dimensional benchmark problems covering a wide range of Knudsen numbers, as well as on a realistic 3D fin field-effect transistor (FinFET) structure. Results demonstrate that MC-PINNs consistently outperform purely data-driven deep neural networks, particularly in the sparse-data regime, and can accurately infer spatially uniform relaxation times. For spatially varying relaxation times, the inferred distributions capture the dominant thermal response, and numerical simulations using the recovered parameters reproduce the macroscopic fields with good accuracy. These findings establish MC-PINNs as an effective and physically consistent framework for inverse thermal analysis at micro- and nanoscales.

This paper presents a gas kinetic Lax-Wendroff scheme (GKLWS)-based finite volume lattice Boltzmann method for high Mach compressible flows. The scheme is constructed based on the total energy kinetic model, which realizes the exact and efficient recovery of the full compressible Navier-Stokes equations through Hermite expansions. A characteristic temperature is introduced to non-dimensionalize the equilibrium states, such that the truncation errors can be reduced and the numerical stability can be enhanced in simulating high Mach compressible problems. Furthermore, the total energy kinetic model is performed on an unstructured second-order accurate finite volume discretization formulation, and the GKLWS is implemented to reconstruct the fluxes on the cell interfaces, enabling the present scheme to compute the compressible flows over complex geometries with lower numerical dissipations. Numerical tests are conducted to validate the present scheme, including compressible Couette flow, the Riemann problem, the double Mach reflection problem, the shock boundary layer interaction and the hypersonic flow over a cylinder. Comparisons with the benchmark results confirm that the proposed scheme is a reliable tool for high Mach compressible flow simulations.

A compact discrete unified gas kinetic scheme (CDUGKS) and its compact steady-state DUGKS (CSDUGKS) are developed in this work to solve a variety of particle-based multiscale transport problems. The core innovation lies in a compact reconstruction scheme, which determines the distribution function at cell interfaces using only local information within a single cell, thereby avoiding the use of data from multiple neighboring cells as in traditional DUGKS formulations. This compact reconstruction significantly enhances the accuracy and robustness of both original unsteady and steady DUGKS methods, particularly for strongly inhomogeneous problems on coarse meshes. The proposed compact reconstruction scheme is highly general and can be readily extended to various numerical limiters by incorporating compact gradients. It is also applicable to other frameworks, such as discrete velocity or ordinate methods and related transport solvers. Several representative particle-based multiscale transport problems—including phonon heat conduction, photon gray radiative transfer, and multigroup neutron transport—are conducted to assess the performance of the proposed CDUGKS and CSDUGKS. Numerical results demonstrate that the compact schemes outperform the original DUGKS and SDUGKS in terms of accuracy, particularly in configurations involving steep gradients, large optical thickness variations, and material discontinuities. Moreover, the compact reconstruction operates entirely within a single cell, making it particularly advantageous for massively parallel implementations. Overall, the proposed CDUGKS and CSDUGKS offer a robust, accurate, and scalable framework for solving complicated multiscale transport problems, and provide a promising basis for future extensions to large-scale practical engineering applications.

Kinetic theory offers a promising alternative to conventional turbulence modelling by providing a mesoscopic perspective that naturally captures non-equilibrium physics such as non-Newtonian effects. In this work, we present an extension and theoretical analysis of the kinetic model for incompressible turbulent flows developed by Chen et al. (Atmosphere, 2023, vol. 14(7), p. 1109), constructed for unbounded flows. The first extension is to reselect a relaxation time such that the turbulent transport coefficients are obtained consistently and better align with well-established turbulence theory. The Chapman-Enskog (CE) analysis of the kinetic model reproduces the linear eddy-viscosity and gradient diffusion models for Reynolds stress and turbulent kinetic energy flux at the first order, and yields nonlinear eddy-viscosity and closure models at the second order. In particular, a previously unreported CE solution for turbulent kinetic energy flux is obtained. The second extension is to enable the model for wall-bounded turbulent flows with preserved near-wall asymptotic behaviours. This involves developing a low-Reynolds-number model incorporating wall damping effects and viscous diffusion, with boundary conditions enabling both viscous sublayer resolution and wall function application. Comprehensive validation against experimental and direct numerical simulation data for turbulent Couette flow demonstrates excellent agreement in predicting mean velocity profiles, skin friction coefficients and Reynolds shear-stress distributions, although the near-wall-normal stress anisotropy is underestimated. The results show that averaged turbulent flow behaves similarly to rarefied-gas flow at finite Knudsen number, capturing non-Newtonian effects beyond linear eddy-viscosity models. This kinetic model provides a physics-based foundation for turbulence modelling with reduced empirical dependence.

External body forces influence the evolution of the distribution function in mesoscopic methods through the particle-velocity-space gradient term in the Boltzmann equation. Simplified forcing models for continuum flows can be categorized into moment-expansion-based and Chapman-Enskog-expansion-based models, while their performance in compressible turbulence remains insufficiently explored. Using the partial internal energy double-distribution-function (DDF) framework, this study addresses that gap. We systematically derive the moment constraints for any forcing model to recover the Navier-Stokes-Fourier equations with arbitrary bulk viscosity for the first time. Within the partial internal energy DDF framework, we novelly propose a moment-expansion-based forcing model for compressible flows, which accounts for the higher-order structure of the distribution function without requiring increased Gauss-Hermite quadrature accuracy. For comparison, we also consider two classical Chapman-Enskog-based models, the He model [He et al., J. Comput. Phys. 146, 282 (1998)], which requires higher-order quadrature, and the Kupershtokh model [Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)], which neglects higher-order structural effects, even though both models satisfy the same velocity moment constraints. All three models yield stable simulations of forced compressible turbulence using partial internal energy DDF-based discrete unified gas kinetic scheme. However, nonlinear interactions lead to observable, though small, differences in turbulence statistics under identical initial conditions. Overall, the results indicate that each model is capable of providing reasonable turbulence statistics.

Hydrodynamic electron transport, in which electrical transport in solids resembles fluid hydrodynamics when momentum-conserving electron-electron scattering dominates, has attracted much attention over the past decade. However, its thermal aspects have received considerably less attention. In this paper, electron transport in a graphene Corbino disk is systematically simulated by solving the stationary Boltzmann transport equation with a dual-relaxation-time Callaway model, where momentum-conserving and momentum-relaxing scatterings are explicitly distinguished. By varying the magnetic field intensity and the scattering rates, the electric charge and heat flux responses are compared across the diffusive-to-hydrodynamic crossover under electric-field or temperature-gradient drives. It is shown that magnetic-field-induced deflection of both fluxes is strongly enhanced in the hydrodynamic regime but nearly suppressed in the diffusive regime. Under electric-field driving, a pronounced temperature rise is observed in the hydrodynamic regime due to reduced dissipation, while the diffusive regime remains nearly isothermal. Under temperature-gradient driving, the deflection is reversed relative to the electric-field case. These findings establish that thermal behaviors could provide a sensitive and independent diagnostic of electron hydrodynamics, with the magnetic field being identified as an effective discriminator between collective and dissipative conduction.

Accurate simulation of micro-scale vapour-liquid phase-change flows is crucial for applications such as micro-electromechanical systems (MEMS). The primary challenge lies in developing a method that can efficiently and uniformly describe the vapour phase, the interface, and the liquid phase. In this paper, a robust discrete unified gas kinetic scheme (DUGKS) is developed for micro-scale non-isothermal vapour-liquid phase-change flows based on a simplified Enskog-Vlasov (E-V) equation. Specifically, the Strang-splitting method is introduced to decouple the complex Vlasov long-range forcing and the dense-gas correction terms into separate sub-equations. Furthermore, a semi-implicit scheme is applied to discretize the sub-equations, and the right-rectangle rule is utilized to replace the standard trapezoidal rule for the characteristic line integration. These tailored numerical treatments ensure the capability of the DUGKS framework to successfully solve the complex E-V equation. The present DUGKS framework enables an effective and unified description of the vapour phase, the liquid phase, and their interface. Numerical validations demonstrate that the method accurately predicts interfacial properties, macroscopic quantity jumps within the Knudsen layer, and phase-change dynamics in problems involving vapour-liquid equilibrium, evaporation into vapour, transient evaporation into metastable vapour and phase separation. The results show excellent agreement with DSMC data and theoretical solutions, while outperforming both the discrete velocity method (DVM) and the lattice Boltzmann method (LBM). The proposed DUGKS thus provides a reliable computational tool for micro-scale vapour-liquid phase-change problems.

A general predictor-corrector discrete unified gas kinetic scheme (PRC_DUGKS) is developed to solve Boltzmann transport models with non-conservative collision operators, including both conservative and non-conservative components. Unlike standard DUGKS formulations that rely on auxiliary distribution functions, PRC_DUGKS directly evolves the original distribution function and incorporates a predictor-corrector strategy to consistently couple transport, conservative and non-conservative collision processes. This proposed formulation eliminates the implicitness introduced by the trapezoidal rule in the time integration and overcomes the limitations of standard DUGKS, where macroscopic variables cannot be directly reconstructed from the moments of the auxiliary distribution function in transport problems with non-conservative collision operators. Furthermore, PRC_DUGKS mitigates the drawbacks of Strang-splitting-based DUGKS (SI_DUGKS), which suffers from operator-splitting errors, accuracy degradation, and numerical instabilities in multiscale problems. The proposed PRC_DUGKS is applied to multigroup neutron transport problems with the conservative and non-conservative collision operators, involving fission, inter-group scattering and absorption terms. Numerical results demonstrate that PRC_DUGKS provides excellent agreement with reference solutions, robustly captures steep-gradient features, and maintains accuracy across regimes with varying optical thicknesses. In contrast, SI_DUGKS exhibits significant discrepancies in regions with strong absorbers and non-conservative interactions, particularly under coarse time discretization. These results demonstrate the superior accuracy and numerical efficiency of PRC_DUGKS, highlighting its potential for large-scale engineering applications involving multiscale transport and complex collision physics.

Neither molecular kinetics nor continuum fluid dynamics alone is adequate to describe multiscale gas flows across different regimes. Bridging these regimes within a single self-consistent framework has long been a central challenge in fluid mechanics. We propose a unified gas kinetic framework that classifies molecules by their collision histories over an observation timescale. This formulation recovers the Boltzmann and Navier–Stokes equations as limiting cases, providing a transparent connection between kinetic and hydrodynamic descriptions. Beyond practical advantages for multiscale modeling, this framework offers a new perspective on Hilbert’s sixth problem by linking microscopic dynamics to continuum mechanics through a tunable observational scale.

In this paper, a microscopically conservation-enforced discrete unified gas kinetic scheme (MicroC-DUGKS) is proposed for multiscale flow simulation. The proposed method rigorously satisfies conservation of mass, momentum, and total energy through preserving the collision term's conservation properties at the discrete level, and ensures accurate heat flux evaluation. Therefore, MicroC-DUGKS achieves accuracy comparable to the original DUGKS with far fewer discrete velocity points. To validate the performance of MicroC-DUGKS, several numerical tests are conducted, including (a) the one-dimensional Lax shock-tube problem, (b) the two-dimensional Riemann problem and (c) supersonic flow around a square cylinder. Numerical tests in the present work demonstrate that: (1) in cases at Knudsen number Kn < 0.1, MicroC-DUGKS reduces memory usage by approximately 70% compared to the original DUGKS while maintaining comparable accuracy; in cases at Knudsen number Kn >= 0.1, MicroC-DUGKS fails to improve accuracy and introduces additional computational cost; (2) in cases at Knudsen number Kn < 0.1, MicroC-DUGKS outperforms both the original DUGKS in heat flux evaluation accuracy when using the same discrete velocity set. Moreover, numerical results indicate that conservative corrections for the entire physical domain are unnecessary. Based on these findings, a criterion for adaptive conservative corrections is provided and preliminary results on adaptive MicroC-DUGKS are presented, suggesting its potential to further enhance computational efficiency.

In this paper, a discrete unified gas kinetic scheme (DUGKS) is developed based on low-Mach-number (LMN) Navier-Stokes-Fourier (NSF) equations for closed systems. The proposed scheme takes advantage of the mesoscopic numerical framework, in which a double distribution function model is adopted to recover the macroscopic continuity, momentum, and energy equations under the LMN approximation. Based on a decoupled formulation of the LMN equations, thermal pressure is separated from momentum evolution. This framework enables the introduction of a numerical Mach number for the discrete velocity set so that the timestep size for the flow simulation is not constrained by the physical speed of sound. Furthermore, the Hermite expansion orders of the equilibrium distribution functions can be reduced for computational efficiency. Two optimized sets of discrete particle velocity models are used, namely, the D2V7A5 velocity set is employed for the g distribution function to simulate dynamic pressure and momentum, and the D2V4A3 set for the h distribution function to recover temperature. These reduced velocity sets are sufficient since higher-order Mach number terms are neglected under the LMN approximation. The separation of dynamic pressure from the thermodynamic pressure and the use of reduced Hermite quadrature orders have significantly improved the computational efficiency. To validate the proposed scheme, we consider several canonical benchmark cases characterized by strong thermal variation and buoyancy-driven flows. The numerical results are compared with the solutions from the fully compressible scheme (FCS) and reference data from the literature. The total speed-up of our proposed scheme is about 557.27 and 8.89, when compared to FCS, with the normalized temperature variation epsilon at 0.01 and 0.6, respectively. In the above speed-up, the use of numerical Mach numbers contributes to a factor of 152.87 and 2.66, respectively.

In this work, a ensemble-of-subproblems strategy with stochastic discrete velocities is extended to deterministic methods for mitigating ray effects in rarefied flow simulations. The strategy involves performing multiple independent subproblems, each using a small set of randomly sampled velocity points, and then averaging their solutions to obtain the final result. The core idea is to ensure that the distribution function at any velocity can contribute to the final result, approximating highly refined velocity-space resolution without increasing the memory requirement in any single subproblem. We incorporate this strategy within the DUGKS framework, and the resulting method is denoted as SDV-DUGKS. To evaluate the performance of the proposed method, we compare SDV-DUGKS with the original DUGKS on several test cases: (a) the Sod shock tube problem, (b) the one-dimensional Riemann problem, (c) the two-dimensional lid-driven cavity flow, and (d) the two-dimensional Riemann problem. The results show that, in the collisionless limit Kn→∞: (1) for one-dimensional compressible flows, SDV-DUGKS reduces memory usage by approximately 2/3 compared with that of the original DUGKS while achieving good agreement; (2) for two-dimensional compressible flows, SDV-DUGKS requires one to two orders of magnitude less memory than the original DUGKS while achieving good agreement. Based on these results, it can be concluded that the proposed method serves as a reliable and effective tool for mitigating ray effects in rarefied flow simulations.

Non-equilibrium evaporative flows play a central role in many nanoporous membrane technologies, where transport of fluids is confined by solid surfaces at the nanoscale. In this work, we propose a molecular kinetic model that consistently resolves the coupled interactions among vapour, liquid and solid surfaces in such flows. As a direct consequence of this bottom-up approach, the liquid-vapour, liquid-solid and vapour-solid interfaces form autonomously, and the effects of non-equilibrium and real fluids can be captured simultaneously, which does not need empirical models depending on ad hoc parameters such as the evaporation/condensation coefficients and contact angle. Accuracy of the model improves further by including the soft-collision effect in the pair correlation function and applying a temperature-dependent correction to the mean-field Vlasov term, as validated against the experimental data and the molecular dynamics simulations. Furthermore, when applied to unsteady, evaporation-driven liquid-vapour flows, the model reveals distinct dynamics due to surface wettability: the hydrophilic surfaces exhibit phenomena such as liquid meniscus breakage and enhanced evaporation flux, whereas the hydrophobic surfaces lead to disappearance of liquid droplets. These findings highlight the potential of the proposed molecular kinetic model as a powerful design tool for next-generation nano-technologies that leverage nano-confined phase change.

The solution of the nonlinear gray radiative transfer system composed of the radiative transfer equation (RTE) and the material energy balance equation encounters significant challenges in multiscale problems where both optical thick and optical thin regions coexist. In this paper, an asymptotic-preserving (AP) scheme based on the discrete unified gas-kinetic scheme (DUGKS) framework is proposed. The microscopic RTE is solved by a simplified DUGKS, where the cell interface intensity is simply constructed from the averaging along the characteristic line, and the macroscopic material energy balance equation is solved by a finite volume method consistent with the DUGKS. The time marching of the present method exhibits the implicit characteristic, and the mesh size is not strictly constrained by the photon mean free path. As a result, the present method has the AP property. Theoretical analysis is conducted to prove the AP property, and several single- and multiscale tests are carried out to confirm the accuracy and robustness of the present method.

In this work, a high-order discrete unified gas kinetic (HDUGKS) scheme for multiscale gas flows was presented. The proposed scheme is a high-order finite volume method, which holds good conservation property of the scheme. Moreover, in the proposed scheme, the high-order accuracy was achieved by a constrained interpolation profile conservative semi-Lagrangian (CIP-CSL) reconstruction, where both point values (PVs) and volume-integrated averages (VIAs) are defined and used to fulfill a compact reconstruction of high order polynomials in a local finite volume cell. The PVs are updated using semi-Lagrangian scheme along the characteristics of the kinetic model equations to maintain high efficiency and low numerical dissipation, but the VIAs are evolved under the discrete unified gas kinetic scheme (DUGKS) framework to preserve the conservation. To demonstrate the idea of the HDUGKS, a cubic polynomial reconstruction based one-dimensional HDUGKS was presented. Furthermore, the high order accuracy property of the proposed scheme was evaluated by five one-dimensional numerical benchmarks including: (a) 1-D sine wave, (b) shock structure, (c) Sod’s shock tube problem, (d) Lax shock tube problem, (e) Shu-Osher problem, and the numerical results show that, while there is one order of accuracy reduction in the proposed scheme due to the low order semi-Lagrangian updating of the PVs, the proposed scheme can almost achieve third order accuracy and obtain more accurate results with fewer meshes than the DUGKS. Additionally, the numerical results indicate that the present method maintains the multiscale properties of the original DUGKS and serve as an efficient numerical method for flow simulations in all Knudsen number flow regime as well.