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Journal articles on the topic 'Discrete Kinetic Scheme'

Author: Grafiati

Published: 6 September 2023

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1

Chandrashekar, Praveen. "Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes for Compressible Euler and Navier-Stokes Equations." Communications in Computational Physics 14, no. 5 (November 2013): 1252–86. http://dx.doi.org/10.4208/cicp.170712.010313a.

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AbstractCentered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the formwhereandareanyconsistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.

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2

Zhu, Lianhua, Zhaoli Guo, and Kun Xu. "Discrete unified gas kinetic scheme on unstructured meshes." Computers & Fluids 127 (March 2016): 211–25. http://dx.doi.org/10.1016/j.compfluid.2016.01.006.

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3

Aregba–Driollet, D., J. Breil, S. Brull, B. Dubroca, and E. Estibals. "Modelling and numerical approximation for the nonconservative bitemperature Euler model." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 4 (July 2018): 1353–83. http://dx.doi.org/10.1051/m2an/2017007.

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This paper is devoted to the study of the nonconservative bitemperature Euler system. We firstly introduce an underlying two species kinetic model coupled with the Poisson equation. The bitemperature Euler system is then established from this kinetic model according to an hydrodynamic limit. A dissipative entropy is proved to exist and a solution is defined to be admissible if it satisfies the related dissipation property. Next, four different numerical methods are presented. Firstly, the kinetic model gives rise to kinetic schemes for the fluid system. The second approach belongs to the family of the discrete BGK schemes introduced by Aregba–Driollet and Natalini. Finally, a quasi-linear relaxation approach and a Lagrange-remap scheme are considered.

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4

Zhong, Mingliang, Sen Zou, Dongxin Pan, Congshan Zhuo, and Chengwen Zhong. "A simplified discrete unified gas–kinetic scheme for compressible flow." Physics of Fluids 33, no. 3 (March 1, 2021): 036103. http://dx.doi.org/10.1063/5.0033911.

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5

Shang, Jinlong, Zhenhua Chai, Xinmeng Chen, and Baochang Shi. "Discrete unified gas kinetic scheme for incompressible Navier-Stokes equations." Computers & Mathematics with Applications 97 (September 2021): 45–60. http://dx.doi.org/10.1016/j.camwa.2021.05.019.

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6

Zhong, Mingliang, Sen Zou, Dongxin Pan, Congshan Zhuo, and Chengwen Zhong. "A simplified discrete unified gas kinetic scheme for incompressible flow." Physics of Fluids 32, no. 9 (September 1, 2020): 093601. http://dx.doi.org/10.1063/5.0021332.

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7

Zhou, Xiafeng, and Zhaoli Guo. "Discrete unified gas kinetic scheme for steady multiscale neutron transport." Journal of Computational Physics 423 (December 2020): 109767. http://dx.doi.org/10.1016/j.jcp.2020.109767.

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8

Wang, Peng, Shi Tao, and Zhaoli Guo. "A coupled discrete unified gas-kinetic scheme for Boussinesq flows." Computers & Fluids 120 (October 2015): 70–81. http://dx.doi.org/10.1016/j.compfluid.2015.07.012.

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9

Guo, Wenqiang, and Guoxiang Hou. "Novel Schemes of No-Slip Boundary Conditions for the Discrete Unified Gas Kinetic Scheme Based on the Moment Constraints." Entropy 25, no. 5 (May 10, 2023): 780. http://dx.doi.org/10.3390/e25050780.

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The boundary conditions are crucial for numerical methods. This study aims to contribute to this growing area of research by exploring boundary conditions for the discrete unified gas kinetic scheme (DUGKS). The importance and originality of this study are that it assesses and validates the novel schemes of the bounce back (BB), non-equilibrium bounce back (NEBB), and Moment-based boundary conditions for the DUGKS, which translate boundary conditions into constraints on the transformed distribution functions at a half time step based on the moment constraints. A theoretical assessment shows that both present NEBB and Moment-based schemes for the DUGKS can implement a no-slip condition at the wall boundary without slip error. The present schemes are validated by numerical simulations of Couette flow, Poiseuille flow, Lid-driven cavity flow, dipole–wall collision, and Rayleigh–Taylor instability. The present schemes of second-order accuracy are more accurate than the original schemes. Both present NEBB and Moment-based schemes are more accurate than the present BB scheme in most cases and have higher computational efficiency than the present BB scheme in the simulation of Couette flow at high Re. The present Moment-based scheme is more accurate than the present BB, NEBB schemes, and reference schemes in the simulation of Poiseuille flow and dipole–wall collision, compared to the analytical solution and reference data. Good agreement with reference data in the numerical simulation of Rayleigh–Taylor instability shows that they are also of use to the multiphase flow. The present Moment-based scheme is more competitive in boundary conditions for the DUGKS.

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10

MIEUSSENS, LUC. "DISCRETE VELOCITY MODEL AND IMPLICIT SCHEME FOR THE BGK EQUATION OF RAREFIED GAS DYNAMICS." Mathematical Models and Methods in Applied Sciences 10, no. 08 (November 2000): 1121–49. http://dx.doi.org/10.1142/s0218202500000562.

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We present a numerical method for computing transitional flows as described by the BGK equation of gas kinetic theory. Using the minimum entropy principle to define a discrete equilibrium function, a discrete velocity model of this equation is proposed. This model, like the continuous one, ensures positivity of solutions, conservation of moments, and dissipation of entropy. The discrete velocity model is then discretized in space and time by an explicit finite volume scheme which is proved to satisfy the previous properties. A linearized implicit scheme is then derived to efficiently compute steady-states; this method is then verified with several test cases.

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11

Sun, Dongke. "A discrete kinetic scheme to model anisotropic liquid–solid phase transitions." Applied Mathematics Letters 103 (May 2020): 106222. http://dx.doi.org/10.1016/j.aml.2020.106222.

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12

Zhang, Chunhua, Kang Yang, and Zhaoli Guo. "A discrete unified gas-kinetic scheme for immiscible two-phase flows." International Journal of Heat and Mass Transfer 126 (November 2018): 1326–36. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2018.06.016.

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13

Yang, Zeren, Sha Liu, Congshan Zhuo, and Chengwen Zhong. "Free-Energy-Based Discrete Unified Gas Kinetic Scheme for van der Waals Fluid." Entropy 24, no. 9 (August 27, 2022): 1202. http://dx.doi.org/10.3390/e24091202.

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The multiphase model based on free-energy theory has been experiencing long-term prosperity for its solid foundation and succinct implementation. To identify the main hindrance to developing a free-energy-based discrete unified gas-kinetic scheme (DUGKS), we introduced the classical lattice Boltzmann free-energy model into the DUGKS implemented with different flux reconstruction schemes. It is found that the force imbalance amplified by the reconstruction errors prevents the direct application of the free-energy model to the DUGKS. By coupling the well-balanced free-energy model with the DUGKS, the influences of the amplified force imbalance are entirely removed. Comparative results demonstrated a consistent performance of the well-balanced DUGKS despite the reconstruction schemes utilized. The capability of the DUGKS coupled with the well-balanced free-energy model was quantitatively validated by the coexisting density curves and Laplace’s law. In the quiescent droplet test, the magnitude of spurious currents is reduced to a machine accuracy of 10−15. Aside from the excellent performance of the well-balanced DUGKS in predicting steady-state multiphase flows, the spinodal decomposition test and the droplet coalescence test revealed its stability problems in dealing with transient flows. Further improvements are required on this point.

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14

Gan, Yanbiao, Aiguo Xu, Guangcai Zhang, and Huilin Lai. "Three-dimensional discrete Boltzmann models for compressible flows in and out of equilibrium." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232, no. 3 (November 29, 2017): 477–90. http://dx.doi.org/10.1177/0954406217742181.

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We present a series of three-dimensional discrete Boltzmann models for compressible flows in and out of equilibrium. The key formulating technique is the construction of discrete equilibrium distribution function through inversely solving the kinetic moment relations that it satisfies. The crucial physical requirement is that all the used kinetic moment relations must be consistent with the non-equilibrium statistical mechanics. The necessity of such a kinetic model is that, with increasing the complexity of flows, the dynamical characterization of non-equilibrium state and the understanding of the constitutive relations need higher order kinetic moments and their evolution. The discrete Boltzmann models at the Euler and Navier–Stokes levels proposed by this scheme are validated by several well-known benchmarks, ranging from one-dimension to three-dimension. Particularly, when the local Mach number, temperature ratio, and pressure ratio are as large as 102, 104, and 105, respectively, the simulation results are still in excellent agreement with the Riemann solutions. How to model deeper thermodynamic non-equilibrium flows by discrete Boltzmann is indicated. Via the discrete Boltzmann method, it is convenient to simulate nonequilibrium flows without knowing exact form of the hydrodynamic equations.

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15

Gan, Yanbiao, Aiguo Xu, Guangcai Zhang, Junqi Wang, Xijun Yu, and Yang Yang. "Lattice Boltzmann kinetic modeling and simulation of thermal liquid–vapor system." International Journal of Modern Physics C 25, no. 12 (December 2014): 1441002. http://dx.doi.org/10.1142/s0129183114410022.

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We present a highly efficient lattice Boltzmann (LB) kinetic model for thermal liquid–vapor system. Three key components are as below: (i) a discrete velocity model (DVM) by Kataoka et al. [Phys. Rev. E69, 035701(R) (2004)]; (ii) a forcing term Ii aiming to describe the interfacial stress and recover the van der Waals (VDW) equation of state (EOS) by Gonnella et al. [Phys. Rev. E76, 036703 (2007)] and (iii) a Windowed Fast Fourier Transform (WFFT) scheme and its inverse by our group [Phys. Rev. E84, 046715 (2011)] for solving the spatial derivatives, together with a second-order Runge–Kutta (RK) finite difference scheme for solving the temporal derivative in the LB equation. The model is verified and validated by well-known benchmark tests. The results recovered from the present model are well consistent with previous ones [Phys. Rev. E84, 046715 (2011)] or theoretical analysis. The usage of less discrete velocities, high-order RK algorithm and WFFT scheme with 16th-order in precision makes the model more efficient by about 10 times and more accurate than the original one.

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16

Leibner, Tobias, and Mario Ohlberger. "A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 6 (November 2021): 2567–608. http://dx.doi.org/10.1051/m2an/2021065.

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In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of solutions. However, their practical use is limited by their high computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.

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17

DU, RUI, and BAOCHANG SHI. "A NOVEL SCHEME FOR FORCE TERM IN THE LATTICE BGK MODEL." International Journal of Modern Physics C 17, no. 07 (July 2006): 945–58. http://dx.doi.org/10.1142/s0129183106009461.

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In this paper a novel scheme of the lattice BGK (LBGK) model with a force term has been proposed. Unlike the existing models, an appropriate term was added in the evolutionary equation. Through the Chapman–Enskog (C–E) procedure the Navier–Stokes (N–S) equations with a force term can be recovered with the kinetic viscosity. Three discrete methods of the added term have been discussed in detail. It can be proved that some existing models are the special cases of the model in this paper. We have taken the numerical simulation of the generalized Poiseuille flow driven by a constant force in a channel filled with a porous medium of porosity flow in 2D with different values of the parameters and compared the three models of different discrete schemes in the aspect of the numerical accuracy and stability.

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18

Aristov, V. V., O. V. Ilyin, and O. A. Rogozin. "Kinetic multiscale scheme based on the discrete-velocity and lattice-Boltzmann methods." Journal of Computational Science 40 (February 2020): 101064. http://dx.doi.org/10.1016/j.jocs.2019.101064.

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19

Arun, K. R., and M. Lukáčová-Medviďová. "A Characteristics Based Genuinely Multidimensional Discrete Kinetic Scheme for the Euler Equations." Journal of Scientific Computing 55, no. 1 (June 28, 2012): 40–64. http://dx.doi.org/10.1007/s10915-012-9623-6.

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20

Huo, Yutao, and Zhonghao Rao. "The discrete unified gas kinetic scheme for solid-liquid phase change problem." International Communications in Heat and Mass Transfer 91 (February 2018): 187–95. http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.12.018.

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21

Song, Xinliang, Yue Zhang, Xiafeng Zhou, Chuang Zhang, and Zhaoli Guo. "Modified steady discrete unified gas kinetic scheme for multiscale radiative heat transfer." International Journal of Heat and Mass Transfer 203 (April 2023): 123799. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2022.123799.

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22

Subbareddy, Pramod K., and Graham V. Candler. "A fully discrete, kinetic energy consistent finite-volume scheme for compressible flows." Journal of Computational Physics 228, no. 5 (March 2009): 1347–64. http://dx.doi.org/10.1016/j.jcp.2008.10.026.

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23

Zhu, Lianhua, and Zhaoli Guo. "Application of discrete unified gas kinetic scheme to thermally induced nonequilibrium flows." Computers & Fluids 193 (October 2019): 103613. http://dx.doi.org/10.1016/j.compfluid.2017.09.019.

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24

Mendoza, M., J. D. Debus, S. Succi, and H. J. Herrmann. "Lattice kinetic scheme for generalized coordinates and curved spaces." International Journal of Modern Physics C 25, no. 12 (December 2014): 1441001. http://dx.doi.org/10.1142/s0129183114410010.

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We present a new lattice kinetic method to simulate fluid dynamics in curvilinear geometries and curved spaces. A suitable discrete Boltzmann equation is solved in contravariant coordinates, and the equilibrium distribution function is obtained by a Hermite polynomials expansion of the Maxwell–Boltzmann distribution, expressed in terms of the contravariant coordinates and the metric tensor. To validate the model, we calculate the critical Reynolds number for the onset of the Taylor–Couette instability between two concentric cylinders, obtaining excellent agreement with the theory. In order to extend this study to more general geometries, we also calculate the critical Reynolds number for the case of two concentric spheres, finding good agreement with experimental data, and the case of two concentric tori, where we have found that it is around 10% larger than the respective values for the two concentric cylinders.

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25

Yang, Zeren, Sha Liu, Congshan Zhuo, and Chengwen Zhong. "Conservative multilevel discrete unified gas kinetic scheme for modeling multiphase flows with large density ratios." Physics of Fluids 34, no. 4 (April 2022): 043316. http://dx.doi.org/10.1063/5.0086723.

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A novel multilevel discrete unified gas kinetic scheme (MDUGKS) is proposed to efficiently model multiphase flows with large density ratios. By constructing the kinetic flux with a mutable time interval depending on the local mesh spacing, the MDUGKS overcomes the stability problems encountered by the standard DUGKS when operating with the multilevel mesh. With the interpolation of macroscopic variables and mesoscopic distributions handled separately, the moments of distribution functions are maintained consistent with the conservative flow variables. Two kinetic equations corresponding to the conservative Allen–Cahn equation and the hydrodynamic equation are individually solved by the MDUGKS, and six benchmark problems have been conducted to evaluate its performance. Numerical solutions in steady cases produced by the MDUGKS are in well accordance with the theoretical predictions. A limiting density ratio of 109 is achieved in the quiescent droplet. The dynamic processes in unsteady cases anticipated by the MDUGKS agree well with the reference predictions. Comparative results also demonstrate that the MDUGKS behaves consistently with different types of meshes. With the employment of the adaptive multilevel mesh, 80% improvement in computational efficiency could be achieved compared with the uniform mesh. Considering the kinetic nature and the high efficiency, the MDUGKS offers a powerful tool for presenting meaningful insight into understanding the realistic multiphase systems at the mesoscopic scale.

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26

Mallet, Jessy, Stéphane Brull, and Bruno Dubroca. "An Entropic Scheme for an Angular Moment Model for the Classical Fokker-Planck-Landau Equation of Electrons." Communications in Computational Physics 15, no. 2 (February 2014): 422–50. http://dx.doi.org/10.4208/cicp.050612.030513a.

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AbstractIn plasma physics domain, the electron transport is described with the Fokker-Planck-Landau equation. The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. That is why we propose in this paper a new model whose derivation is based on an angular closure in the phase space and retains only the energy of particles as kinetic dimension. To find a solution compatible with physics conditions, the closure of the moment system is obtained under a minimum entropy principle. This model is proved to satisfy the fundamental properties like a H theorem. Moreover an entropic discretization in the velocity variable is proposed on the semi-discrete model. Finally, we validate on numerical test cases the fundamental properties of the full discrete model.

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27

Herbin, Raphaèle, Jean-Claude Latché, and Chady Zaza. "A cell-centred pressure-correction scheme for the compressible Euler equations." IMA Journal of Numerical Analysis 40, no. 3 (June 17, 2019): 1792–837. http://dx.doi.org/10.1093/imanum/drz024.

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Abstract We propose a robust pressure-correction scheme for the numerical solution of the compressible Euler equations discretized by a collocated finite volume method. The scheme is based on an internal energy formulation, which ensures that the internal energy is positive. More generally, the scheme enjoys fundamental stability properties: without restriction on the time step, both the density and the internal energy are positive, the integral of the total energy over the computational domain is preserved thanks to an estimate on the discrete kinetic energy and a discrete entropy inequality is satisfied. These stability properties ensure the existence of a solution to the scheme. The internal energy balance features a corrective source term, which is needed for the scheme to compute the correct shock solutions; we are indeed able to prove a Lax-consistency-type convergence result, in the sense that, under some compactness assumptions, the limit of a converging sequence of approximate solutions obtained with space and time discretization steps tending to zero is an entropy weak solution of the Euler equations. Moreover, constant pressure and velocity are preserved through contact discontinuities. The obtained theoretical results and the scheme accuracy are verified by numerical experiments; a numerical stabilization is introduced in order to reduce the oscillations that appear for some tests. The qualitative behaviour of the scheme is assessed on one-dimensional and two-dimensional Riemann problems and compared with other schemes.

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28

Zhan, Ningyu, Rongqian Chen, and Yancheng You. "Meshfree method based on discrete gas-kinetic scheme to simulate incompressible/compressible flows." Physics of Fluids 33, no. 1 (January 1, 2021): 017112. http://dx.doi.org/10.1063/5.0033770.

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29

Wen, Xin, Lian-Ping Wang, Zhaoli Guo, and Jie Shen. "An improved discrete unified gas kinetic scheme for simulating compressible natural convection flows." Journal of Computational Physics: X 11 (June 2021): 100088. http://dx.doi.org/10.1016/j.jcpx.2021.100088.

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30

Zhan, Ningyu, Rongqian Chen, and Yancheng You. "Discrete gas-kinetic scheme-based arbitrary Lagrangian–Eulerian method for moving boundary problems." Physics of Fluids 33, no. 6 (June 2021): 067101. http://dx.doi.org/10.1063/5.0051299.

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31

Wu, Chen, Baochang Shi, Zhenhua Chai, and Peng Wang. "Discrete unified gas kinetic scheme with a force term for incompressible fluid flows." Computers & Mathematics with Applications 71, no. 12 (June 2016): 2608–29. http://dx.doi.org/10.1016/j.camwa.2016.04.025.

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32

Zhao, Xiang, Chen Wu, Zhen Chen, Liming Yang, and Chang Shu. "Reduced order modeling-based discrete unified gas kinetic scheme for rarefied gas flows." Physics of Fluids 32, no. 6 (June 1, 2020): 067108. http://dx.doi.org/10.1063/5.0009614.

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33

Zhang, Chuang, and Zhaoli Guo. "Discrete unified gas kinetic scheme for multiscale heat transfer with arbitrary temperature difference." International Journal of Heat and Mass Transfer 134 (May 2019): 1127–36. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2019.02.056.

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34

Diaz, Manuel A., Min-Hung Chen, and Jaw-Yen Yang. "High-Order Conservative Asymptotic-Preserving Schemes for Modeling Rarefied Gas Dynamical Flows with Boltzmann-BGK Equation." Communications in Computational Physics 18, no. 4 (October 2015): 1012–49. http://dx.doi.org/10.4208/cicp.171214.210715s.

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AbstractHigh-order and conservative phase space direct solvers that preserve the Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas flows are explored and studied. The approach is based on the conservative discrete ordinate method for velocity space by using Gauss Hermite or Simpsons quadrature rule and conservation of macroscopic properties are enforced on the BGK collision operator. High-order asymptotic-preserving time integration is adopted and the spatial evolution is performed by high-order schemes including a finite difference weighted essentially non-oscillatory method and correction procedure via reconstruction schemes. An artificial viscosity dissipative model is introduced into the Boltzmann-BGK equation when the correction procedure via reconstruction scheme is used. The effects of the discrete velocity conservative property and accuracy of high-order formulations of kinetic schemes based on BGK model methods are provided. Extensive comparative tests with one-dimensional and two-dimensional problems in rarefied gas flows have been carried out to validate and illustrate the schemes presented. Potentially advantageous schemes in terms of stable large time step allowed and higher-order of accuracy are suggested.

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35

Hochbruck, Marlis, and Jan Leibold. "An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions." Numerische Mathematik 147, no. 4 (March 3, 2021): 869–99. http://dx.doi.org/10.1007/s00211-021-01184-w.

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AbstractWe construct and analyze a second-order implicit–explicit (IMEX) scheme for the time integration of semilinear second-order wave equations. The scheme treats the stiff linear part of the problem implicitly and the nonlinear part explicitly. This makes the scheme unconditionally stable and at the same time very efficient, since it only requires the solution of one linear system of equations per time step. For the combination of the IMEX scheme with a general, abstract, nonconforming space discretization we prove a full discretization error bound. We then apply the method to a nonconforming finite element discretization of an acoustic wave equation with a kinetic boundary condition. This yields a fully discrete scheme and a corresponding a-priori error estimate.

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36

Zhu, Yajun, Chengwen Zhong, and Kun Xu. "GKS and UGKS for High-Speed Flows." Aerospace 8, no. 5 (May 19, 2021): 141. http://dx.doi.org/10.3390/aerospace8050141.

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The gas-kinetic scheme (GKS) and the unified gas-kinetic scheme (UGKS) are numerical methods based on the gas-kinetic theory, which have been widely used in the numerical simulations of high-speed and non-equilibrium flows. Both methods employ a multiscale flux function constructed from the integral solutions of kinetic equations to describe the local evolution process of particles’ free transport and collision. The accumulating effect of particles’ collision during transport process within a time step is used in the construction of the schemes, and the intrinsic simulating flow physics in the schemes depends on the ratio of the particle collision time and the time step, i.e., the so-called cell’s Knudsen number. With the initial distribution function reconstructed from the Chapman–Enskog expansion, the GKS can recover the Navier–Stokes solutions in the continuum regime at a small Knudsen number, and gain multi-dimensional properties by taking into account both normal and tangential flow variations in the flux function. By employing a discrete velocity distribution function, the UGKS can capture highly non-equilibrium physics, and is capable of simulating continuum and rarefied flow in all Knudsen number regimes. For high-speed non-equilibrium flow simulation, the real gas effects should be considered, and the computational efficiency and robustness of the schemes are the great challenges. Therefore, many efforts have been made to improve the validity and reliability of the GKS and UGKS in both the physical modeling and numerical techniques. In this paper, we give a review of the development of the GKS and UGKS in the past decades, such as physical modeling of a diatomic gas with molecular rotation and vibration at high temperature, plasma physics, computational techniques including implicit and multigrid acceleration, memory reduction methods, and wave–particle adaptation.

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37

He, Qing, Shi Tao, Xiaoping Yang, Weijian Lu, and Zongrun He. "Discrete unified gas kinetic scheme simulation of microflows with complex geometries in Cartesian grid." Physics of Fluids 33, no. 4 (April 2021): 042005. http://dx.doi.org/10.1063/5.0040850.

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38

Liu, Peiyao. "A Coupled Discrete Unified Gas-Kinetic Scheme for Convection Heat Transfer in Porous Media." Communications in Computational Physics 29, no. 1 (June 2021): 265–91. http://dx.doi.org/10.4208/cicp.oa-2019-0200.

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39

Guo, Wenqiang, and Guoxiang Hou. "Three-Dimensional Simulations of Anisotropic Slip Microflows Using the Discrete Unified Gas Kinetic Scheme." Entropy 24, no. 7 (June 30, 2022): 907. http://dx.doi.org/10.3390/e24070907.

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The specific objective of the present work study is to propose an anisotropic slip boundary condition for three-dimensional (3D) simulations with adjustable streamwise and spanwise slip length by the discrete unified gas kinetic scheme (DUGKS). The present boundary condition is proposed based on the assumption of nonlinear velocity profiles near the wall instead of linear velocity profiles in a unidirectional steady flow. Moreover, a 3D corner boundary condition is introduced to the DUGKS to reduce the singularities. Numerical tests validate the effectiveness of the present method, which is more accurate than the bounce-back and specular reflection slip boundary condition in the lattice Boltzmann method. It is of significance to study the lid-driven cavity flow due to its applications and its capability in exhibiting important phenomena. Then, the present work explores, for the first time, the effects of anisotropic slip on the two-sided orthogonal oscillating micro-lid-driven cavity flow by adopting the present method. This work will generate fresh insight into the effects of anisotropic slip on the 3D flow in a two-sided orthogonal oscillating micro-lid-driven cavity. Some findings are obtained: The oscillating velocity of the wall has a weaker influence on the normal velocity component than on the tangential velocity component. In most cases, large slip length has a more significant influence on velocity profiles than small slip length. Compared with pure slip in both top and bottom walls, anisotropic slip on the top wall has a greater influence on flow, increasing the 3D mixing of flow. In short, the influence of slip on the flow field depends not only on slip length but also on the relative direction of the wall motion and the slip velocity. The findings can help in better understanding the anisotropic slip effect on the unsteady microflow and the design of microdevices.

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Tao, Shi, Haolong Zhang, Zhaoli Guo, and Lian-Ping Wang. "A combined immersed boundary and discrete unified gas kinetic scheme for particle–fluid flows." Journal of Computational Physics 375 (December 2018): 498–518. http://dx.doi.org/10.1016/j.jcp.2018.08.047.

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Zhang, L. Q., Z. Chen, L. M. Yang, and C. Shu. "Double distribution function-based discrete gas kinetic scheme for viscous incompressible and compressible flows." Journal of Computational Physics 412 (July 2020): 109428. http://dx.doi.org/10.1016/j.jcp.2020.109428.

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Lee, Wook, Eunbeom Jung, Seongwon Kang, and Nahmkeon Hur. "On a momentum interpolation scheme for collocated meshes with improved discrete kinetic energy conservation." Journal of Mechanical Science and Technology 33, no. 6 (June 2019): 2761–68. http://dx.doi.org/10.1007/s12206-019-0522-8.

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Tao, Shi, Baiman Chen, Xiaoping Yang, and Simin Huang. "Second-order accurate immersed boundary-discrete unified gas kinetic scheme for fluid-particle flows." Computers & Fluids 165 (March 2018): 54–63. http://dx.doi.org/10.1016/j.compfluid.2018.01.005.

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Marcos, Aboubacar, and Ambroise Soglo. "Solutions of a Class of Degenerate Kinetic Equations Using Steepest Descent in Wasserstein Space." Journal of Mathematics 2020 (June 9, 2020): 1–30. http://dx.doi.org/10.1155/2020/7489532.

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We use the steepest descent method in an Orlicz–Wasserstein space to study the existence of solutions for a very broad class of kinetic equations, which include the Boltzmann equation, the Vlasov–Poisson equation, the porous medium equation, and the parabolic p-Laplacian equation, among others. We combine a splitting technique along with an iterative variational scheme to build a discrete solution which converges to a weak solution of our problem.

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Bhatt, Maulik, Amit K. Sanyal, and Srikant Sukumar. "Asymptotically stable optimal multi-rate rigid body attitude estimation based on lagrange-d'alembert principle." Journal of Geometric Mechanics 15, no. 1 (2023): 73–97. http://dx.doi.org/10.3934/jgm.2023004.

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<abstract><p>The rigid body attitude estimation problem is treated using the discrete-time Lagrange-d'Alembert principle. Three different possibilities are considered for the multi-rate relation between angular velocity measurements and direction vector measurements for attitude: 1) integer relation between sampling rates, 2) time-varying sampling rates, 3) non-integer relation between sampling rates. In all cases, it is assumed that angular velocity measurements are sampled at a higher rate compared to the inertial vectors. The attitude determination problem from two or more vector measurements in the body-fixed frame is formulated as Wahba's problem. At instants when direction vector measurements are absent, a discrete-time model for attitude kinematics is used to propagate past measurements. A discrete-time Lagrangian is constructed as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential energy-like term obtained from Wahba's cost function. An additional dissipation term is introduced and the discrete-time Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation to obtain an optimal filtering scheme. A discrete-time Lyapunov analysis is carried out to show that the optimal filtering scheme is asymptotically stable in the absence of measurement noise and the domain of convergence is almost global. For a realistic evaluation of the scheme, numerical experiments are conducted with inputs corrupted by bounded measurement noise. These numerical simulations exhibit convergence of the estimated states to a bounded neighborhood of the actual states.</p></abstract>

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Allgeyer, Sebastien, Marie-Odile Bristeau, David Froger, Raouf Hamouda, V. Jauzein, Anne Mangeney, Jacques Sainte-Marie, Fabien Souillé, and Martin Vallée. "Numerical approximation of the 3D hydrostatic Navier–Stokes system with free surface." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 6 (November 2019): 1981–2024. http://dx.doi.org/10.1051/m2an/2019044.

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In this paper we propose a stable and robust strategy to approximate the 3D incompressible hydrostatic Euler and Navier–Stokes systems with free surface. Compared to shallow water approximation of the Navier–Stokes system, the idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a strategy has several advantages. It allows to rewrite the Navier–Stokes equations under the form of a system of conservation laws with source terms,the easy handling of the free surface, which does not require moving meshes,the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems. Compared to previous works of some of the authors, the three dimensional case is studied in this paper. We show that the model admits a kinetic interpretation including the vertical exchanges terms, and we use this result to formulate a robust finite volume scheme for its numerical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions, ...) are completely described and the stability properties of the proposed numerical scheme (well-balancing, positivity of the water depth, ...) are discussed. We validate the model and the discrete scheme with some numerical academic examples (3D non stationary analytical solutions) and illustrate the capability of the discrete model to reproduce realistic tsunami waves propagation, tsunami runup and complex 3D hydrodynamics in a raceway.

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Yang, L. M., C. Shu, Y. Wang, and Y. Sun. "Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows." Journal of Computational Physics 319 (August 2016): 129–44. http://dx.doi.org/10.1016/j.jcp.2016.05.018.

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Chattopadhyay, Somesh, Daniel M. Keenan, and Johannes D. Veldhuis. "Probabilistic recovery of neuroendocrine pulsatile, secretory and kinetic structure: An alternating discrete and continuous scheme." Quarterly of Applied Mathematics 66, no. 3 (March 18, 2008): 401–21. http://dx.doi.org/10.1090/s0033-569x-08-01024-4.

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Li, Chao, and Lian-Ping Wang. "An immersed boundary-discrete unified gas kinetic scheme for simulating natural convection involving curved surfaces." International Journal of Heat and Mass Transfer 126 (November 2018): 1059–70. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2018.04.166.

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Liu, Hongtao, Yong Cao, Qing Chen, Mingchi Kong, and Liang Zheng. "A conserved discrete unified gas kinetic scheme for microchannel gas flows in all flow regimes." Computers & Fluids 167 (May 2018): 313–23. http://dx.doi.org/10.1016/j.compfluid.2018.03.023.

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