Close Search Advanced
Journals
Resources
About Us
Open Access

Shock and Boundary Structure Formation by Spectral-Lagrangian Methods for the Inhomogeneous Boltzmann Transport Equation

Shock and Boundary Structure Formation by Spectral-Lagrangian Methods for the Inhomogeneous Boltzmann Transport Equation
Preview
Add to basket

Year:    2010

Journal of Computational Mathematics, Vol. 28 (2010), Iss. 4 : pp. 430–460

Abstract

The numerical approximation of the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock and boundary layer structures, we present the computational algorithm in Section 3.3 and perform a numerical study case in shock tube geometries well modeled in for $1D$ in $\textbf{x}$ times $3D$ in $\textbf{v}$ in Section 4. The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Aoki et al [2], where the wall temperature is suddenly increased or decreased, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting in this example is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock.

Submit Article

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/jcm.1003-m0011

Journal of Computational Mathematics, Vol. 28 (2010), Iss. 4 : pp. 430–460

Published online:    2010-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    31

Keywords:    Spectral Numerical Methods Lagrangian optimization FFT Boltzmann Transport Equation Conservative and non-conservative rarefied gas flows.

  1. An efficient numerical method for solving the Boltzmann equation in multidimensions

    Dimarco, Giacomo | Loubère, Raphaël | Narski, Jacek | Rey, Thomas

    Journal of Computational Physics, Vol. 353 (2018), Iss. P.46

    https://doi.org/10.1016/j.jcp.2017.10.010 [Citations: 49]

  2. Spectral method for a kinetic swarming model

    Gamba, Irene M. | Haack, Jeffrey R. | Motsch, Sebastien

    Journal of Computational Physics, Vol. 297 (2015), Iss. P.32

    https://doi.org/10.1016/j.jcp.2015.04.033 [Citations: 12]

  3. High order modal Discontinuous Galerkin Implicit–Explicit Runge Kutta and Linear Multistep schemes for the Boltzmann model on general polygonal meshes

    Boscheri, Walter | Dimarco, Giacomo

    Computers & Fluids, Vol. 233 (2022), Iss. P.105224

    https://doi.org/10.1016/j.compfluid.2021.105224 [Citations: 7]

  4. High order finite volume schemes with IMEX time stepping for the Boltzmann model on unstructured meshes

    Boscheri, Walter | Dimarco, Giacomo

    Computer Methods in Applied Mechanics and Engineering, Vol. 387 (2021), Iss. P.114180

    https://doi.org/10.1016/j.cma.2021.114180 [Citations: 11]

  5. A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

    Gamba, Irene M. | Haack, Jeffrey R.

    Journal of Computational Physics, Vol. 270 (2014), Iss. P.40

    https://doi.org/10.1016/j.jcp.2014.03.035 [Citations: 23]

  6. Fast evaluation of the Boltzmann collision operator using data driven reduced order models

    Alekseenko, Alexander | Martin, Robert | Wood, Aihua

    Journal of Computational Physics, Vol. 470 (2022), Iss. P.111526

    https://doi.org/10.1016/j.jcp.2022.111526 [Citations: 4]

  7. A Fast Spectral Method for the Boltzmann Collision Operator with General Collision Kernels

    Gamba, Irene M. | Haack, Jeffrey R. | Hauck, Cory D. | Hu, Jingwei

    SIAM Journal on Scientific Computing, Vol. 39 (2017), Iss. 4 P.B658

    https://doi.org/10.1137/16M1096001 [Citations: 59]

  8. Spectral computation of low probability tails for the homogeneous Boltzmann equation

    Zweck, John | Chen, Yanping | Goeckner, Matthew J. | Shen, Yannan

    Applied Numerical Mathematics, Vol. 162 (2021), Iss. P.301

    https://doi.org/10.1016/j.apnum.2020.12.027 [Citations: 0]

  9. Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space

    Alekseenko, A. | Josyula, E.

    Journal of Computational Physics, Vol. 272 (2014), Iss. P.170

    https://doi.org/10.1016/j.jcp.2014.03.031 [Citations: 27]

  10. Convergence and Error Estimates for the Lagrangian-Based Conservative Spectral Method for Boltzmann Equations

    Alonso, Ricardo J. | Gamba, Irene M. | Tharkabhushanam, Sri Harsha

    SIAM Journal on Numerical Analysis, Vol. 56 (2018), Iss. 6 P.3534

    https://doi.org/10.1137/18M1173332 [Citations: 11]

  11. A Polynomial Spectral Method for the Spatially Homogeneous Boltzmann Equation

    Kitzler, Gerhard | Schöberl, Joachim

    SIAM Journal on Scientific Computing, Vol. 41 (2019), Iss. 1 P.B27

    https://doi.org/10.1137/17M1160240 [Citations: 7]
  12. Handbook of Numerical Methods for Hyperbolic Problems - Applied and Modern Issues

    Deterministic Solvers for Nonlinear Collisional Kinetic Flows

    Gamba, I.M.

    2017

    https://doi.org/10.1016/bs.hna.2016.11.005 [Citations: 0]

  13. A rescaling velocity method for dissipative kinetic equations. Applications to granular media

    Filbet, Francis | Rey, Thomas

    Journal of Computational Physics, Vol. 248 (2013), Iss. P.177

    https://doi.org/10.1016/j.jcp.2013.04.023 [Citations: 21]

  14. A Fast Petrov--Galerkin Spectral Method for the Multidimensional Boltzmann Equation Using Mapped Chebyshev Functions

    Hu, Jingwei | Huang, Xiaodong | Shen, Jie | Yang, Haizhao

    SIAM Journal on Scientific Computing, Vol. 44 (2022), Iss. 3 P.A1497

    https://doi.org/10.1137/21M1420721 [Citations: 3]
  15. Uncertainty Quantification for Hyperbolic and Kinetic Equations

    Uncertainty Quantification for Kinetic Equations

    Hu, Jingwei | Jin, Shi

    2017

    https://doi.org/10.1007/978-3-319-67110-9_6 [Citations: 12]

  16. Numerical methods for kinetic equations

    Dimarco, G. | Pareschi, L.

    Acta Numerica, Vol. 23 (2014), Iss. P.369

    https://doi.org/10.1017/S0962492914000063 [Citations: 253]

  17. Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

    Holloway, Ian | Wood, Aihua | Alekseenko, Alexander

    Mathematics, Vol. 9 (2021), Iss. 12 P.1384

    https://doi.org/10.3390/math9121384 [Citations: 5]

  18. Method of calculating the collision integral and solution of the Boltzmann kinetic equation for simple gases, gas mixtures and gases with rotational degrees of freedom

    Anikin, Yu.A. | Dodulad, O.I. | Kloss, Yu.Yu. | Tcheremissine, F.G.

    International Journal of Computer Mathematics, Vol. 92 (2015), Iss. 9 P.1775

    https://doi.org/10.1080/00207160.2014.909033 [Citations: 9]

  19. A spectral-Lagrangian Boltzmann solver for a multi-energy level gas

    Munafò, Alessandro | Haack, Jeffrey R. | Gamba, Irene M. | Magin, Thierry E.

    Journal of Computational Physics, Vol. 264 (2014), Iss. P.152

    https://doi.org/10.1016/j.jcp.2014.01.036 [Citations: 28]

  20. A stochastic Galerkin method for the Boltzmann equation with uncertainty

    Hu, Jingwei | Jin, Shi

    Journal of Computational Physics, Vol. 315 (2016), Iss. P.150

    https://doi.org/10.1016/j.jcp.2016.03.047 [Citations: 53]

  21. Deterministic numerical solutions of the Boltzmann equation using the fast spectral method

    Wu, Lei | White, Craig | Scanlon, Thomas J. | Reese, Jason M. | Zhang, Yonghao

    Journal of Computational Physics, Vol. 250 (2013), Iss. P.27

    https://doi.org/10.1016/j.jcp.2013.05.003 [Citations: 113]

  22. Moment Preserving Fourier–Galerkin Spectral Methods and Application to the Boltzmann Equation

    Pareschi, Lorenzo | Rey, Thomas

    SIAM Journal on Numerical Analysis, Vol. 60 (2022), Iss. 6 P.3216

    https://doi.org/10.1137/21M1423452 [Citations: 7]

  23. A Bhatnagar–Gross–Krook kinetic model with velocity-dependent collision frequency and corrected relaxation of moments

    Alekseenko, Alexander | Euler, Craig

    Continuum Mechanics and Thermodynamics, Vol. 28 (2016), Iss. 3 P.751

    https://doi.org/10.1007/s00161-014-0407-0 [Citations: 2]

  24. Gain of integrability for the Boltzmann collisional operator

    J. Alonso, Ricardo | M. Gamba, Irene

    Kinetic & Related Models, Vol. 4 (2011), Iss. 1 P.41

    https://doi.org/10.3934/krm.2011.4.41 [Citations: 12]

  25. Galerkin–Petrov approach for the Boltzmann equation

    Gamba, Irene M. | Rjasanow, Sergej

    Journal of Computational Physics, Vol. 366 (2018), Iss. P.341

    https://doi.org/10.1016/j.jcp.2018.04.017 [Citations: 28]

  26. A Conservative Discontinuous Galerkin Solver for the Space Homogeneous Boltzmann Equation for Binary Interactions

    Zhang, Chenglong | Gamba, Irene M.

    SIAM Journal on Numerical Analysis, Vol. 56 (2018), Iss. 5 P.3040

    https://doi.org/10.1137/16M1104792 [Citations: 6]

  27. Implicit gas-kinetic unified algorithm based on multi-block docking grid for multi-body reentry flows covering all flow regimes

    Peng, Ao-Ping | Li, Zhi-Hui | Wu, Jun-Lin | Jiang, Xin-Yu

    Journal of Computational Physics, Vol. 327 (2016), Iss. P.919

    https://doi.org/10.1016/j.jcp.2016.09.050 [Citations: 54]

  28. Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation

    Fonn, E. | Grohs, P. | Hiptmair, R.

    IMA Journal of Numerical Analysis, Vol. 35 (2015), Iss. 4 P.1533

    https://doi.org/10.1093/imanum/dru042 [Citations: 4]

  29. On Deterministic Approximation of the Boltzmann Equation in a Bounded Domain

    Filbet, Francis

    Multiscale Modeling & Simulation, Vol. 10 (2012), Iss. 3 P.792

    https://doi.org/10.1137/11082419X [Citations: 18]

  30. Investigation of nonequilibrium effects across normal shock waves by means of a spectral-Lagrangian Boltzmann solver

    Munafò, Alessandro | Torres, Erik | Haack, Jeff | Gamba, Irene | Magin, Thierry

    51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, (2013),

    https://doi.org/10.2514/6.2013-305 [Citations: 0]