Journals
Resources
About Us
Open Access

A Well-Balanced Kinetic Scheme for Gas Dynamic Equations under Gravitational Field

A Well-Balanced Kinetic Scheme for Gas Dynamic Equations under Gravitational Field

Year:    2010

Author:    Kun Xu, Jun Luo, Songze Chen

Advances in Applied Mathematics and Mechanics, Vol. 2 (2010), Iss. 2 : pp. 200–210

Abstract

In this paper, a well-balanced kinetic scheme for the gas dynamic equations under gravitational field is developed. In order to construct such a scheme, the physical process of particles transport through a potential barrier at a cell interface is considered, where the amount of particle penetration and reflection is evaluated according to the incident particle velocity. This work extends the approach of Perthame and Simeoni for the shallow water equations [Calcolo, 38 (2001), pp. 201-231] to the Euler equations under gravitational field. For an isolated system, this scheme is probably the only well-balanced method which can precisely preserve an isothermal steady state solution under time-independent gravitational potential. A few numerical examples are used to validate the above approach.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m0964

Advances in Applied Mathematics and Mechanics, Vol. 2 (2010), Iss. 2 : pp. 200–210

Published online:    2010-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    11

Keywords:    Gas-kinetic scheme Gas-kinetic scheme

Author Details

Kun Xu

Jun Luo

Songze Chen

  1. A well-balanced unified gas-kinetic scheme for multiscale flow transport under gravitational field

    Xiao, Tianbai | Cai, Qingdong | Xu, Kun

    Journal of Computational Physics, Vol. 332 (2017), Iss. P.475

    https://doi.org/10.1016/j.jcp.2016.12.022 [Citations: 21]
  2. Treating network junctions in finite volume solution of transient gas flow models

    Bermúdez, Alfredo | López, Xián | Vázquez-Cendón, M. Elena

    Journal of Computational Physics, Vol. 344 (2017), Iss. P.187

    https://doi.org/10.1016/j.jcp.2017.04.066 [Citations: 19]
  3. High order well-balanced conservative finite difference AWENO scheme with hydrostatic reconstruction for the Euler equations under gravitational fields

    Fu, Qingcheng | Gao, Zhen | Gu, Yaguang | Li, Peng

    Applied Numerical Mathematics, Vol. 180 (2022), Iss. P.1

    https://doi.org/10.1016/j.apnum.2022.05.005 [Citations: 2]
  4. A Comparative Analysis of Fractional-Order Gas Dynamics Equations via Analytical Techniques

    Zhou, Shuang-Shuang | Shah, Nehad Ali | Dassios, Ioannis | Saleem, S. | Nonlaopon, Kamsing

    Mathematics, Vol. 9 (2021), Iss. 15 P.1735

    https://doi.org/10.3390/math9151735 [Citations: 2]
  5. Validation of a two-dimensional gas-kinetic scheme for compressible natural convection on structured and unstructured meshes

    Lenz, Stephan | Krafczyk, Manfred | Geier, Martin | Chen, Songze | Guo, Zhaoli

    International Journal of Thermal Sciences, Vol. 136 (2019), Iss. P.299

    https://doi.org/10.1016/j.ijthermalsci.2018.10.004 [Citations: 9]
  6. A well-balanced scheme for the shallow-water equations with topography or Manning friction

    Michel-Dansac, Victor | Berthon, Christophe | Clain, Stéphane | Foucher, Françoise

    Journal of Computational Physics, Vol. 335 (2017), Iss. P.115

    https://doi.org/10.1016/j.jcp.2017.01.009 [Citations: 28]
  7. Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry

    Zhang, Weijie | Xing, Yulong | Endeve, Eirik

    Monthly Notices of the Royal Astronomical Society, Vol. 514 (2022), Iss. 1 P.370

    https://doi.org/10.1093/mnras/stac1257 [Citations: 2]
  8. A Well-Balanced Symplecticity-Preserving Gas-Kinetic Scheme for Hydrodynamic Equations under Gravitational Field

    Luo, Jun | Xu, Kun | Liu, Na

    SIAM Journal on Scientific Computing, Vol. 33 (2011), Iss. 5 P.2356

    https://doi.org/10.1137/100803699 [Citations: 41]
  9. A Very Easy High-Order Well-Balanced Reconstruction for Hyperbolic Systems with Source Terms

    Berthon, Christophe | Bulteau, Solène | Foucher, Françoise | M'Baye, Meissa | Michel-Dansac, Victor

    SIAM Journal on Scientific Computing, Vol. 44 (2022), Iss. 4 P.A2506

    https://doi.org/10.1137/21M1429230 [Citations: 6]
  10. High order well-balanced discontinuous Galerkin methods for shallow water flow under temperature fields

    Qian, Shouguo | Shao, Fengjing | Li, Gang

    Computational and Applied Mathematics, Vol. 37 (2018), Iss. 5 P.5775

    https://doi.org/10.1007/s40314-018-0662-y [Citations: 8]
  11. A Moving Mesh Method for Kinetic/Hydrodynamic Coupling

    Hu, Zhicheng | Wang, Heyu

    Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 06 P.685

    https://doi.org/10.4208/aamm.12-12S01 [Citations: 2]
  12. Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines

    Bermúdez, Alfredo | López, Xián | Vázquez-Cendón, M. Elena

    Journal of Computational Physics, Vol. 323 (2016), Iss. P.126

    https://doi.org/10.1016/j.jcp.2016.07.020 [Citations: 24]
  13. High order finite volume WENO schemes for the Euler equations under gravitational fields

    Li, Gang | Xing, Yulong

    Journal of Computational Physics, Vol. 316 (2016), Iss. P.145

    https://doi.org/10.1016/j.jcp.2016.04.015 [Citations: 38]
  14. A well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity

    Desveaux, Vivien | Zenk, Markus | Berthon, Christophe | Klingenberg, Christian

    International Journal for Numerical Methods in Fluids, Vol. 81 (2016), Iss. 2 P.104

    https://doi.org/10.1002/fld.4177 [Citations: 38]
  15. Well-Balanced Discontinuous Galerkin Methods for the Euler Equations Under Gravitational Fields

    Li, Gang | Xing, Yulong

    Journal of Scientific Computing, Vol. 67 (2016), Iss. 2 P.493

    https://doi.org/10.1007/s10915-015-0093-5 [Citations: 29]
  16. On high order positivity-preserving well-balanced finite volume methods for the Euler equations with gravitation

    Ren, Yupeng | Wu, Kailiang | Qiu, Jianxian | Xing, Yulong

    Journal of Computational Physics, Vol. 492 (2023), Iss. P.112429

    https://doi.org/10.1016/j.jcp.2023.112429 [Citations: 1]
  17. Theory, Numerics and Applications of Hyperbolic Problems I

    Well-Balanced Central-Upwind Schemes for $$2\,\times \,2$$ Systems of Balance Laws

    Chertock, Alina | Herty, Michael | Özcan, Şeyma Nur

    2018

    https://doi.org/10.1007/978-3-319-91545-6_28 [Citations: 11]
  18. A well-balanced scheme for Ten-Moment Gaussian closure equations with source term

    Meena, Asha Kumari | Kumar, Harish

    Zeitschrift für angewandte Mathematik und Physik, Vol. 69 (2018), Iss. 1

    https://doi.org/10.1007/s00033-017-0901-x [Citations: 7]
  19. Well-balanced finite difference WENO schemes for the Ripa model

    Han, Xiao | Li, Gang

    Computers & Fluids, Vol. 134-135 (2016), Iss. P.1

    https://doi.org/10.1016/j.compfluid.2016.04.031 [Citations: 17]
  20. A Well-Balanced Discontinuous Galerkin Method Based on Hydrostatic Reconstruction for Real Gas in Pipelines

    郭, 威

    Advances in Applied Mathematics, Vol. 10 (2021), Iss. 12 P.4404

    https://doi.org/10.12677/AAM.2021.1012469 [Citations: 0]
  21. Well-balanced methods for computational astrophysics

    Käppeli, Roger

    Living Reviews in Computational Astrophysics, Vol. 8 (2022), Iss. 1

    https://doi.org/10.1007/s41115-022-00014-6 [Citations: 6]
  22. Simple high order well-balanced finite difference WENO schemes for the Euler equations under gravitational fields

    Li, Peng | Gao, Zhen

    Journal of Computational Physics, Vol. 437 (2021), Iss. P.110341

    https://doi.org/10.1016/j.jcp.2021.110341 [Citations: 10]
  23. Well‐balanced discontinuous Galerkin method and finite volume WENO scheme based on hydrostatic reconstruction for blood flow model in arteries

    Li, Gang | Delestre, Olivier | Yuan, Li

    International Journal for Numerical Methods in Fluids, Vol. 86 (2018), Iss. 7 P.491

    https://doi.org/10.1002/fld.4463 [Citations: 12]
  24. Well‐balanced finite difference weighted essentially non‐oscillatory schemes for the blood flow model

    Wang, Zhenzhen | Li, Gang | Delestre, Olivier

    International Journal for Numerical Methods in Fluids, Vol. 82 (2016), Iss. 9 P.607

    https://doi.org/10.1002/fld.4232 [Citations: 14]
  25. High order well-balanced positivity-preserving scale-invariant AWENO scheme for Euler systems with gravitational field

    Gu, Yaguang | Gao, Zhen | Hu, Guanghui | Li, Peng | Fu, Qingcheng

    Journal of Computational Physics, Vol. 488 (2023), Iss. P.112190

    https://doi.org/10.1016/j.jcp.2023.112190 [Citations: 3]
  26. Uniformly High-Order Structure-Preserving Discontinuous Galerkin Methods for Euler Equations with Gravitation: Positivity and Well-Balancedness

    Wu, Kailiang | Xing, Yulong

    SIAM Journal on Scientific Computing, Vol. 43 (2021), Iss. 1 P.A472

    https://doi.org/10.1137/20M133782X [Citations: 18]
  27. A well-balanced Runge-Kutta discontinuous Galerkin method for the Euler equations in isothermal hydrostatic state under gravitational field

    Chen, Ziming | Zhang, Yingjuan | Li, Gang | Qian, Shouguo

    Computers & Mathematics with Applications, Vol. 119 (2022), Iss. P.340

    https://doi.org/10.1016/j.camwa.2022.05.025 [Citations: 1]
  28. Well-balanced discontinuous Galerkin methods with hydrostatic reconstruction for the Euler equations with gravitation

    Li, Gang | Xing, Yulong

    Journal of Computational Physics, Vol. 352 (2018), Iss. P.445

    https://doi.org/10.1016/j.jcp.2017.09.063 [Citations: 37]
  29. Sensitivity Parameter-Independent Characteristic-Wise Well-Balanced Finite Volume WENO Scheme for the Euler Equations Under Gravitational Fields

    Li, Peng | Wang, Bao-Shan | Don, Wai-Sun

    Journal of Scientific Computing, Vol. 88 (2021), Iss. 2

    https://doi.org/10.1007/s10915-021-01562-4 [Citations: 8]
  30. Well-balanced schemes for the Euler equations with gravitation: Conservative formulation using global fluxes

    Chertock, Alina | Cui, Shumo | Kurganov, Alexander | Özcan, Şeyma Nur | Tadmor, Eitan

    Journal of Computational Physics, Vol. 358 (2018), Iss. P.36

    https://doi.org/10.1016/j.jcp.2017.12.026 [Citations: 61]
  31. High Order Well-Balanced WENO Scheme for the Gas Dynamics Equations Under Gravitational Fields

    Xing, Yulong | Shu, Chi-Wang

    Journal of Scientific Computing, Vol. 54 (2013), Iss. 2-3 P.645

    https://doi.org/10.1007/s10915-012-9585-8 [Citations: 78]
  32. High order well-balanced discontinuous Galerkin methods for Euler equations at isentropic equilibrium state under gravitational fields

    Qian, Shouguo | Liu, Yu | Li, Gang | Yuan, Li

    Applied Mathematics and Computation, Vol. 329 (2018), Iss. P.23

    https://doi.org/10.1016/j.amc.2018.01.059 [Citations: 2]
  33. Positivity-preserving well-balanced central discontinuous Galerkin schemes for the Euler equations under gravitational fields

    Jiang, Haili | Tang, Huazhong | Wu, Kailiang

    Journal of Computational Physics, Vol. 463 (2022), Iss. P.111297

    https://doi.org/10.1016/j.jcp.2022.111297 [Citations: 4]