This paper presents the development of a well-balanced gas-kinetic scheme (GKS) with space-time adaptive mesh refinement (STAMR) for the shallow water equations (SWE). While well-balanced GKS have been established on Cartesian and triangular meshes, the proposed STAMR framework utilizes arbitrary quadrilateral meshes with hanging nodes, introducing additional challenges for maintaining well-balanced properties. In addition to spatial adaptivity, temporal adaptivity is incorporated by assigning adaptive time steps to cells at different refinement levels, further enhancing computational efficiency. Furthermore, the numerical flux in the GKS adaptively transitions between equilibrium fluxes for smooth flows and non-equilibrium fluxes for discontinuities, providing the proposed GKS-based STAMR method with strong robustness, high accuracy, and high resolution. Standard benchmark tests and real-world case studies validate the effectiveness of the GKS-based STAMR and demonstrate its potential for interface capturing and the simulation of complex flows.

Simulating multiscale flows with moving boundaries, such as hypersonic multi-body separation and flows in micro-electro-mechanical systems (MEMS), requires robust numerical methods that couple mesh deformation with complex flow physics. This paper presents a hybrid overlapping moving-mesh technique developed within the unified gas-kinetic scheme (UGKS). To mitigate the Courant-Friedrichs-Lewy (CFL) constraint, we extend the implicit unsteady UGKS solver to support moving meshes, incorporating memory-efficient data handling and parallel computing optimizations to maximize computational efficiency. Validated against hypersonic multi-body separation and thermal rarefied MEMS flows, the proposed scheme accurately resolves complex, dynamic multiscale phenomena. The results confirm that this robust and efficient method provides a highly reliable tool for modeling dynamic flow interactions in complex geometric configurations.

Fast and accurate fluid simulation of radio-frequency capacitively coupled plasmas (RF CCPs) is of great importance for the iterative design and parameter optimization of modern plasma reactors. This study presents the first successful extension of the time-domain harmonic balance (HB) method to a fully coupled drift-diffusion-Poisson system with complete electron-energy transport for RF plasma simulations. To resolve the severe numerical stiffness arising from highly nonlinear energy-dependent kinetics and dense phase-coupling, a highly efficient spatiotemporal operator-splitting strategy is employed. By sequentially executing a spatial implicit relaxation and a cell-local temporal inversion, this strategy entirely avoids the memory-intensive assembly of global Jacobians while preserving robust numerical stability. The proposed method is rigorously validated against a standard parallel-plate argon CCP benchmark. Evaluated across all discrete temporal collocation points, the HB solution demonstrates that retaining eight harmonics perfectly resolves both the quasi-steady bulk plasma and the highly nonlinear transient sheath dynamics, yielding macroscopic relative errors strictly below 0.3

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 paper introduces a memory-reduction third-order compact gas-kinetic scheme (CGKS) for solving compressible Euler and Navier-Stokes equations on 3D unstructured meshes. The scheme utilizes a time-evolution gas distribution function to provide a time-evolution solution at cell interfaces, enabling the implementation of Hermite WENO techniques for high-order reconstruction. However, the HWENO method needs to store a coefficients matrix for the quadratic polynomial to achieve third-order accuracy, resulting in high memory usage. A novel reconstruction method, built upon HWENO reconstruction, has been designed to enhance computational efficiency and reduce memory usage compared to the original CGKS. The simple idea is that the first-order and second-order terms of the quadratic polynomials are determined in a two-step way. In the first step, the second-order terms are obtained from the reconstruction of a linear polynomial of the first-order derivatives by only using the cell-averaged slopes, since the second-order derivatives are nothing but the "derivatives of derivatives". Subsequently, the first-order terms left can be determined by the linear reconstruction only using cell-averaged values. Thus, we successfully split one quadratic least-square regression into several linear least-square regressions, which are commonly used in a second-order finite volume code. Since only a small matrix inversion is needed in a 3-D linear least-square regression, the computational cost for the new reconstruction is dramatically reduced and the storage of the reconstruction-coefficient matrix is no longer necessary. The proposed new reconstruction technique can reduce the overall computational cost by about 20 to 30 percent. The challenging large-scale unsteady numerical simulation is performed, which demonstrates that the current improvement brings the CGKS to a new level for industrial applications.

This study presents an efficient and accurate discrete adjoint gas-kinetic scheme (GKS) for sensitivity analysis and aerodynamic shape optimization in continuum flow regimes. Developed using the backward mode of algorithmic differentiation (AD), the adjoint solver is rigorously verified against a duality-preserving linearized GKS solver generated via forward-mode AD. The robustness and practical effectiveness of the solver are evaluated through three benchmark cases: the inverse design of turbine blades, lift-to-drag ratio enhancement, and shock-strength reduction for a NACA 0012 airfoil. To capture realistic flow physics, fully turbulent optimizations are conducted using the one-equation Spalart–Allmaras (SA) model. Numerical results demonstrate excellent agreement between the discrete adjoint and linearized solvers, exhibiting matching sensitivity convergence behaviors, identical asymptotic residual decay rates, and negligible discrepancies in final sensitivity predictions. Furthermore, the optimization studies confirm that targeted design objectives are consistently achieved within a limited number of design cycles, highlighting the solver's computational efficiency, accuracy, and suitability for complex aerodynamic geometries.

Rarefied gas effects are of critical importance for the aerodynamic performance of hypersonic vehicles operating at high altitudes. In these scenarios, conventional computational fluid dynamics (CFD) solvers break down as the linear constitutive relations underlying the Navier-Stokes equations cease to be valid. Based on direct modeling, the unified gas-kinetic scheme (UGKS) and the unified gas-kinetic wave-particle (UGKWP) method successfully capture non-equilibrium physics across all Knudsen numbers, yet they incur substantially higher computational costs than continuum solvers. Within the same kinetic framework, the gas-kinetic scheme (GKS) employs the Chapman-Enskog expansion for near-equilibrium flow physics and adopts the same kinetic boundary conditions as UGKS and UGKWP. This formulation naturally permits velocity slip and temperature jump, thereby extending the applicability of GKS into the slip and transitional regimes. By utilizing this natural kinetic slip boundary condition, the GKS provides a more physically faithful representation of non-equilibrium wall interactions than conventional CFD solvers equipped with Maxwell-type slip conditions, ultimately yielding more accurate aerodynamic predictions. To determine the applicability of the GKS in near-continuum flow regimes, we first examine a simple circular cylinder geometry, comparing surface quantities and distribution functions in detail. Furthermore, we investigate a 9°blunted cone, a 70° blunted cone with a cylindrical sting, and the Apollo 6 command module. This analysis focuses on integrated aerodynamic predictions, which are validated against experimental data, Direct Simulation Monte Carlo (DSMC) simulations, and other kinetic methods.

A novel fifth-order compact gas-kinetic scheme is developed for high-resolution simulation of compressible flows on structured meshes. Its accuracy relies on a new multidimensional fifth-order compact reconstruction that uses line-averaged derivatives to introduce additional degrees of freedom, enabling a compact stencil with superior resolution. For non-orthogonal meshes, reconstruction is performed on a standard reference cell in a transformed computational space. This approach provides a unified polynomial form, significantly reducing memory usage and computational cost while simplifying implementation compared to direct multi-dimensional or dimension-by-dimension methods. A nonlinear adaptive method ensures high accuracy and robustness by smoothly transitioning from the high-order linear scheme in smooth regions to a second-order scheme at discontinuities. The method is implemented with multi-GPU parallelization using CUDA and MPI for large-scale applications. Comprehensive numerical tests, from subsonic to supersonic turbulence, validate the scheme's high accuracy, resolution and excellent robustness.

This study proposes a well-balanced formulation of weakly compressible smoothed particle hydrodynamics (WCSPH) for free-surface flows, which preserves hydrostatic equilibrium exactly at the discrete level–a property essential for reliable long-term simulations. Although well-balanced schemes are well established for mesh-based methods, the property remains largely unaddressed in WCSPH, where the particle approximation of the pressure gradient fails to balance the gravitational force exactly. The imbalance stems from two difficulties: the nonlinearity of the pressure-gradient-over-density term, and the approximation error of gradients evaluated by particle summation. The first is resolved by introducing an auxiliary potential variable that recasts the nonlinear term as the gradient of a single scalar, which reduces to a linear function of position under hydrostatic conditions. The second is resolved by a Riemann-based gradient approximation with kernel correction, which is first-order consistent and recovers linear fields exactly. These two ingredients ensure that the discrete potential gradient balances gravitational force exactly. Widely used techniques, including δ-SPH, particle shifting and tensile instability control, are readily incorporated. The formulation is further extended to three dimensions and implemented on GPU with architecture-tailored optimizations. Hydrostatic tests with rectangular, triangular and Gaussian bottom topographies show that the proposed formulation attains the well-balanced property to machine precision, reducing the spurious velocity error of conventional SPH from 10^-3 to the order of 10^-13. More complex benchmarks confirm its robustness, accuracy and low pressure oscillation, with simulations of up to 17.53 million particles performed on a single consumer-grade GPU.

The unified gas-kinetic wave-particle (UGKWP) method, developed for the multiscale simulation of partially ionised plasmas, is applied to model electromagnetic flows around a hemisphere spanning regimes from near-continuum to rarefied conditions. To the best of our knowledge, this study presents the first application of a multiscale plasma solver to such a problem. In the formulation, neutrals, ions and electrons are treated as distinct species. The numerical implementation is validated through comparisons with experimental data for a Mach 4.75 pre-ionised argon flow, where UGKWP results show close agreement with the experimental data. A further comparative study across different Knudsen numbers demonstrates that rarefaction effects weaken the influence of electromagnetic control. These findings highlight the capability of UGKWP in modelling electromagnetic control problems and underscore the significant role of rarefied effects in predicting flow-control behaviour, thereby emphasising the necessity of multiscale modelling in plasma flow applications.

The deterministic unified gas-kinetic scheme (UGKS) provides a multiscale framework for nonequilibrium gas dynamics, but its high-dimensional phase-space discretization leads to severe memory pressure and communication overhead, especially on large unstructured meshes. This paper presents a GPGPU-oriented UGKS with velocity-block pipelining and full phase-space MPI decomposition. In the proposed formulation, the discrete velocity space is partitioned into fixed-size velocity blocks for accelerator execution, while MPI ranks are organized into coupled physical-space and velocity-space communicators. As a result, each rank stores and advances only a local physical subdomain together with a contiguous subset of velocity blocks, and macroscopic moments are recovered through lightweight reductions over the velocity-space communicator. To improve concurrency and reduce exposed communication cost, a triple-buffered pipeline is further developed to overlap microscopic reconstruction, physical-halo exchange, nonequilibrium flux evaluation, and the first-stage distribution update during the local velocity-block sweep. The implementation targets SIMT-based GPGPU accelerators through a portable device-runtime abstraction. Numerical experiments demonstrate that the $P_v=8$ configuration achieves a $33.4$--$35.4\times$ strong-scaling speedup on 64 nodes, while an Orion-like capsule simulation reaches approximately $1.33\times10^{11}$ phase-space degrees of freedom on 4096 GPGPU accelerators. These results indicate that the proposed method preserves the original UGKS flux construction and two-stage time discretization, while substantially reducing microscopic storage per rank and improving the scalability of large unstructured phase-space simulations.

The viscous shock tube is a canonical test case for assessing Navier-Stokes (NS) solvers in the continuum-flow regime, widely used to validate numerical accuracy and probe flow physics. It features a rich set of interacting structures-shock and rarefaction waves, contact discontinuities, boundary layers, and their coupling-spanning multiple spatial and temporal scales. However, NS-based modeling, which presumes near-equilibrium behavior, may fail to capture important non-equilibrium effects even in nominally continuum conditions. This study investigates the viscous shock tube at low Reynolds numbers and demonstrates the presence of non-equilibrium phenomena within the conventional continuum regime. To obtain physically consistent solutions across scales, we employ the unified gas-kinetic scheme (UGKS) and compare its results with NS solutions computed using the gas-kinetic scheme (GKS). Discrepancies between UGKS and GKS solutions reveal pronounced non-equilibrium effects in regions where shock waves interact with boundary layers. For continuum flows at high Mach and low Reynolds numbers, such multiscale non-equilibrium transport becomes important, underscoring the need for multiscale methods in analysis and prediction.

The gas-kinetic scheme (GKS) provides high computational efficiency and accuracy for continuum flow simulations but is unable to reliably capture rarefaction effects. In contrast, although the discrete velocity method (DVM) is better suited for rarefied flows, it exhibits reduced accuracy and slow convergence when applied to continuum regimes. To overcome these limitations, this work proposes a hybrid GKS-DVM method that integrates the strengths of both approaches. The hybrid approach balances the equilibrium distribution function in GKS with the upwind-reconstructed non-equilibrium distribution function in DVM through a numerical collision time. This balancing strategy ensures to recover Navier-Stokes solutions in the continuum limit (asymptotic preserving), while naturally capturing free molecular flows in the rarefied limit. Moreover, the introduction of a numerical collision time significantly enhances robustness in shock capturing for continuum flow applications. To further reduce computational cost of the hybrid approach, several adaptive strategies based on the local Knudsen number and Mach number have been proposed. The effectiveness and accuracy of the proposed hybrid method are systematically assessed through three representative test cases: a flat-plate boundary layer, a lid-driven cavity flow, and shock-structure problems. The first case is subjected to continuum conditions, while the latter two span a broad range of Knudsen numbers. The results demonstrate that the proposed method achieves high solution accuracy and computational efficiency across both continuum and rarefied flow regimes.

This study extends the Unified Gas-Kinetic Scheme (UGKS) and the Unified Gas-Kinetic Wave-Particle (UGKWP) method for electrostatic plasma modeling, ensuring the correct asymptotic limits with respect to both the Debye length and the mean free path. By coupling collision and transport processes within the numerical flux, the proposed approach effectively removes the hydrodynamic-limit constraint associated with the mean free path. In addition, a reformulated Poisson equation, coupled with the macroscopic moment equations, is introduced to overcome the inefficiency of the standard Poisson formulation in the quasineutral regime. The accuracy and asymptotic consistency of the proposed schemes are verified through several benchmark tests, including linear and nonlinear Landau damping and the bump-on-tail instability. The results demonstrate that the methods robustly capture plasma dynamics across hydrodynamic and quasineutral regimes, without resolution constraints imposed by either the Debye length or the mean free path.

In this paper, the wave-particle turbulent simulation (WPTS), a recently developed multiscale, non-equilibrium turbulence modeling approach, is coupled with a turbulence characteristic-time closure derived from Prandtl mixing-length hypothesis and applied to spatially developing round jets. In WPTS, fluid elements in strongly turbulent regions are represented by Lagrangian particles that travel a finite distance before interacting with the background flow field represented in a wave-like (Eulerian) form. This mechanism bears conceptual similarity to the discrete fluid parcels invoked in the Prandtl mixing-length picture. WPTS differs from conventional mixing-length-based turbulence models in two key respects. First, particle evolution follows a non-equilibrium transport mechanism, rather than the equilibrium assumptions typically embedded in eddy-viscosity closures. Second, WPTS advances the wave and particle components in a coupled manner, with the particle fraction governed primarily by the modeled turbulence characteristic time, enabling laminar and turbulent regimes to be represented within a unified framework. Because spatially developing jets provide a canonical test case with well-established similarity behavior, they are used here for evaluation. Specifically, this work (1) develops a mixing-length-based characteristic-time model tailored to jet flows and (2) incorporates it into WPTS to assess predictive performance. The resulting WPTS framework accurately reproduces the jet similarity solution and other characteristic features at Reynolds numbers of 5,000 and 20,000, demonstrating the promise of WPTS as a practical tool for turbulence modeling and simulation.

The gas-kinetic scheme (GKS) integrates the characteristics of flux difference scheme (FDS) and flux vector splitting (FVS) scheme, providing high accuracy in smooth regions and strong robustness near discontinuities across all Mach regimes. Leveraging these properties, an implicit discrete adjoint GKS is developed for aerodynamic shape optimization over a wide range of Mach numbers. The adjoint solver is constructed using the source-transformation-based algorithmic differentiation tool Tapenade. To enhance computational efficiency, both the flow and adjoint GKS equations are solved using an implicit time-marching strategy, also known as the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) method. The effectiveness of the implicit formulation is demonstrated through comparisons with the explicit approach. To accurately impose solid wall boundary conditions, particularly in hypersonic regimes, kinetic boundary conditions and their adjoint counterparts are formulated for both adiabatic no-slip and isothermal walls. Four benchmark test cases covering subsonic, transonic, supersonic, and hypersonic flows are used to verify the effectiveness of the developed adjoint-based design optimization system.

Benefiting from its direct modeling of physical laws in discretized space and its automatic decomposition of hydrodynamic waves and particles, the unified gas-kinetic wave-particle (UGKWP) method provides significant advantages for a wide range of multiscale physical problems, including hypersonic flow, plasma transport, and radiation transport. To achieve a more effective and efficient wave-particle decomposition in high-speed flow simulations, particularly in regions with drastic scale variations, this work investigates a scale-adaptive criterion and introduces modifications to the flux evolution of the UGKWP method. In addition to the intrinsic time-based criterion embedded in the time-dependent distribution function of UGKWP, two further criteria-based on spatial resolution and local gradients-are employed to identify the local scale and reduce the computational overhead of particles in representing near-equilibrium gas distributions. Furthermore, by aligning the evolution of hydrodynamic waves with the coefficients in the time-integration flux of the unified gas-kinetic scheme (UGKS), the modified wave representation improves consistency with particle contributions, which is especially critical when flow scales vary significantly across computational cells. The effectiveness of the adaptive UGKWP method is demonstrated through a series of benchmark cases, including hypersonic flows around a cylinder at various inflow Knudsen numbers, hypersonic flow over a slender cavity, side-jet impingement in hypersonic flow, and three-dimensional hypersonic flows over a 70 degrees blunted cone with a cylindrical sting.

Local time stepping (LTS) can accelerate convergence to steady states in kinetic simulations with large variations in the local time steps across the computational domain. When neighboring cells advance with unequal time steps, the time-averaged particle flux must be balanced across their common interface. For particle-based or hybrid wave-particle methods under finite volume method (FVM) framework, we rigorously establish a sufficient condition for time-averaged interfacial particle-flux balance: fixed positive local time steps together with proportional particle-mass scaling. When a particle crosses from cell L to cell R, its mass is scaled by Δt_R/Δt_L. In the unified gas-kinetic wave-particle (UGKWP) implementation, the same ratio is applied to the remaining free-transport time of the crossing particle. LTS also affects the wave-particle decomposition and the time integration of the wave fluxes in UGKWP. The wave-particle decomposition in cell i is determined by the local ratio Δt_i/τ_i, which better reflects the local relation between the observation scale and relaxation time for multiscale cases. The equilibrium and analytic free transport wave fluxes are integrated over and normalized by the corresponding cell-side time steps to obtain the interfacial time-averaged wave fluxes. The UGKWP-LTS method is used to simulate the hypersonic flow past a cylinder at Kn=0.01 and 0.1, and a flat plate at Kn=0.0169. In all three cases, the surface quantities obtained with UGKWP-LTS agree well with the reference data. Relative to global time stepping (GTS), UGKWP-LTS achieves step-count speedups of 6.6×, 3.8×, and 20× for the three cases, respectively. The corresponding wall-clock speedups are 7.1×, 4.5×, and approximately 21×.

Benefiting from the direct modeling of physical laws in a discretized space and the automatic decomposition of the gas distribution function into hydrodynamic waves and particles, the UGKWP method offers significant advantages for multiscale flows such as hypersonic flows, plasma transport, and radiation transport. In this study, the particle sampling accuracy in the UGKWP method is improved from first order to second order, so that the second-order spatial and temporal accuracy is preserved across the full scheme. Specifically, the modifications include second-order particle sampling based on local macroscopic gradients, a weighted least-squares gradient reconstruction that incorporates wall values, a revised Venkatakrishnan limiter for highly stretched cells, and conservation corrections after particle sampling. Moreover, the first-order Chapman–Enskog term is considered in the free-transport part of the hydrodynamic wave flux, enabling better recovery of the GKS in the near-continuum regime. Based on these improvements, the mesh-independence behavior of the UGKWP method is notably enhanced, which is more consistent with the performance of the UGKS, validated by a detailed hypersonic cylinder flow test case. Furthermore, systematic comparisons with the single-scale DSMC method are performed for two-dimensional hypersonic flow over a cylinder and three-dimensional flow over a blunt cone. Wall pressure, shear stress, and heat flux coefficients (CP, CF, and CQ) are examined in the cylinder case, while the overall aerodynamic coefficients (CL, CD, and L/D) are assessed in the cone case. The multiscale UGKWP method exhibits significantly better mesh-independence performance than DSMC for mesh-sensitive quantities such as CF, CQ, CD, and L/D, which are critical for aerodynamic and thermal protection design of near-space hypersonic vehicles.