A Conservative Multi-Level Adaptive Velocity-Space Method for the Discrete Unified Gas-Kinetic Scheme

A multi-level velocity-space adaptive discrete unified gas-kinetic scheme has been developed for the efficient simulation of multiscale nonequilibrium gas flows because it dynamically identifies the velocity-space requirements of individual physical cells and groups them into a limited number of globally shared representative velocity spaces.

Methodology for Velocity Space Adaptation

The method first identifies the local velocity-space requirements and then clusters similar requirements into a limited number of globally shared representative velocity spaces. Each physical cell is assigned one representative space according to its local distribution-function characteristics, which retains local adaptivity through multiple representative spaces while avoiding the need to construct and manage a distinct velocity space for every physical cell. The velocity discretization is based on a quadtree-structured reference grid Vbase.

Moment Consistency and Corrections

A linear moment-constrained correction is incorporated to maintain discrete compatibility and conservative data transfer between representative velocity spaces. This correction is introduced because the analytically constructed collision-model distribution does not necessarily satisfy the prescribed compatibility conditions exactly under discrete quadrature. The eight-moment correction is extended to eight constraints to preserve the selected stress- and heat-flux-related moments.

Performance Gains in Computational Cost

Benchmark simulations spanning near-continuum to highly rarefied regimes show close agreement with the uniform-velocity-space DUGKS (UVS-DUGKS). For the computationally demanding cases, the average number of discrete velocity points is reduced by approximately one order of magnitude, yielding speedups of 6.67–7.51 and GPU-memory reductions by factors of 6.15–6.88. The MLVS-DUGKS achieves speedups of 1.80–2.00 and reduces the GPU-memory requirement by factors of 6.15–6.74 for the Mach-5 cylinder flow.

Validation Across Flow Regimes

The accuracy and adaptivity of the method have been assessed using the Sod shock tube, lid-driven cavity, Mach-5 cylinder flow, and transient gas expansion between two connected cavities. Across the investigated Knudsen-number range, the MLVS-DUGKS remains in close agreement with the UVSDUGKS in the principal macroscopic fields while adapting the representative velocity spaces to shifted, broadened, asymmetric, and multicomponent distribution functions.

Conclusion on Strategy Effectiveness

The threshold study demonstrates that the multi-level organization can outperform a single globally shared adaptive velocity space, achieving a 17.3% runtime reduction and a 6.5% memory reduction in the cylinder case at Kn = 0.01. The present single-GPU implementation was developed primarily to establish and validate the numerical framework and has not yet been extensively optimized for a specific GPU architecture. Future work will focus on GPU-oriented data organization, kernel fusion, reduction of global synchronization, and optimization of mixed-level interface operations.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12202490 and 12502275), the Hunan Province Natural Science Foundation (Grant No. 2024JJ6455), and the Hunan Provincial Innovation Foundation for Postgraduate (Grant Nos. CX20250007 and XJJCS26). The authors gratefully acknowledge Professor Zhaoli Guo of Huazhong University of Science and Technology for his support and guidance throughout this research.

References

[1] Z. Li, Y. Liang, A. Peng, J. Wu, H. Wei, Gas-kinetic unified algorithm for aerodynamics covering various flow regimes by computable modeling of Boltzmann equation

[2] F. Li, Z. Li, J. Hu, W. Hu, Application of Gas-Kinetic Unified Algorithm for Solving Boltzmann Model Equation with Vibrational Energy Excitation

[3] C. E. Glass, Numerically Simulating an Expanding Continuum Jet into a Surrounding Noncontinuum Region

[4] Q. Chen, J. Chen, Y. Zhang, X. Yuan, Numerical simulation of influence of RCS on control characteristics of aerodynamic surface

[5] G. Karniadakis, A. Be¸sk¨ok, N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation

[6] H. Akhlaghi, E. Roohi, S. Stefanov, A new iterative wall heat flux specifying technique in DSMC for heating/cooling simulations of MEMS/NEMS

[7] Q. Li, Y. L. He, G. H. Tang, W. Q. Tao, Lattice Boltzmann modeling of microchannel flows in the transition flow regime

[8] A. Sharma, S. Namsani, J. K Singh Molecular simulation of shale gas adsorption and diffusion in inorganic nanopores

[9] X. Li, Z.-m. Fang, Current status and technical challenges of CO2 storage in coal seams and enhanced coalbed methane recovery: An overview

[10] E. Ozdemir, Modeling of coal bed methane (CBM) production and CO2 sequestration in coal seams

[11] G. A. Bird, The DSMC Method, 1st Edition

[12] G. A. Bird, Approach to translational equilibrium in a rigid sphere gas

[13] C. Liu, Y. Zhu, K Xu, Unified gas-kinetic wave-particle methods I: Continuum and rarefied gas flow

[14] Y. Zhu, C. Liu, C Zhong, K Xu Unified gas-kinetic wave-particle methods II Multiscale simulation on unstructured mesh

[15] K. Xu, J Huang, A unified gas-kinetic scheme for continuum and rarefied flows

[16] J.-C. Huang, K Xu, P Yu A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II Multi-Dimensional Cases

[17] Z. Guo, K Xu, R Wang Discrete unified gas kinetic scheme for all Knudsen number flows: Lowspeed isothermal case

[18] Z. Guo, R Wang, K Xu Discrete unified gas kinetic scheme for all Knudsen number flows II Thermal compressible case

[19] R. Wang, Unified gas-kinetic scheme for the study of non-equilibrium flows Ph.D thesis The Hong Kong University of Science and Technology, Hong Kong

[20] Z. Guo, K Xu, Progress of discrete unified gas-kinetic scheme for multiscale flows Advances in Aerodynamics 3 (1) (2021) 6

[21] H. Liu, Y Cao, G Lapenta R Keppens Discrete unified gas kinetic scheme for multiscale plasma simulations Journal of Computational Physics 559 (2026) 1

[22] R Zhang, C Zhong S Liu C Zhuo Large-eddy simulation of wall-bounded turbulent flow with high-order discrete unified gas-kinetic scheme Advances in Aerodynamics 2 (1) (2020) 6

[23] Y Bo, P Wang Z Guo L.-P. Wang DUGKS simulations of three-dimensional Taylor–Green vortex flow and turbulent channel flow Computers & Fluids 155 (2017) 9–21

[24] P Wang L.-P.

1. Multi-level velocity space adaptation allows AI systems to efficiently simulate multiscale gas flows across wide Knudsen numbers by dynamically grouping local distribution function requirements into a limited number of globally shared representative velocity spaces, effectively reducing velocity space redundancy while preserving accuracy and conservation.

2. Linear moment constrained correction ensures that macroscopic variables and nonequilibrium transport quantities are preserved during distribution function mapping between different velocity spaces, providing a method to enforce the discrete compatibility of the collision-model distribution without iteration.

3. The nine moment correction enables AI systems to preserve higher-order nonequilibrium moments such as stress and heat flux exactly, which is crucial for accurately modeling complex flow phenomena like those encountered in shock waves or boundary layers.