Numerical Study of Partially Conservative Moment Equations in Kinetic Theory

Numerical Study of Partially Conservative Moment Equations in Kinetic Theory

Year:    2017

Communications in Computational Physics, Vol. 21 (2017), Iss. 4 : pp. 981–1011

Abstract

Moment models are often used for the solution of kinetic equations such as the Boltzmann equation. Unfortunately, standard models like Grad's equations are not hyperbolic and can lead to nonphysical solutions. Newly derived moment models like the Hyperbolic Moment Equations and the Quadrature-Based Moment Equations yield globally hyperbolic equations but are given in partially conservative form that cannot be written as a conservative system.
In this paper we investigate the applicability of different dedicated numerical schemes to solve the partially conservative model equations. Caused by the non-conservative type of equation we obtain differences in the numerical solutions, but due to the structure of the moment systems we show that these effects are very small for standard simulation cases. After successful identification of useful numerical settings we show a convergence study for a shock tube problem and compare the results to a discrete velocity solution. The results are in good agreement with the reference solution and we see convergence considering an increasing number of moments.

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/cicp.OA-2016-0053

Communications in Computational Physics, Vol. 21 (2017), Iss. 4 : pp. 981–1011

Published online:    2017-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    31

Keywords:   

  1. Model order reduction for the 1D Boltzmann-BGK equation: identifying intrinsic variables using neural networks

    Koellermeier, Julian | Krah, Philipp | Reiss, Julius | Schellin, Zachary

    Microfluidics and Nanofluidics, Vol. 28 (2024), Iss. 3

    https://doi.org/10.1007/s10404-024-02711-5 [Citations: 0]
  2. Efficient moment method for modeling nanoporous evaporation

    De Fraja, Thomas C. | Rana, Anirudh S. | Enright, Ryan | Cooper, Laura J. | Lockerby, Duncan A. | Sprittles, James E.

    Physical Review Fluids, Vol. 7 (2022), Iss. 2

    https://doi.org/10.1103/PhysRevFluids.7.024201 [Citations: 7]
  3. On the efficient implementation of PVM methods and simple Riemann solvers. Application to the Roe method for large hyperbolic systems

    Pimentel-García, Ernesto | Parés, Carlos | Castro, Manuel J. | Koellermeier, Julian

    Applied Mathematics and Computation, Vol. 388 (2021), Iss. P.125544

    https://doi.org/10.1016/j.amc.2020.125544 [Citations: 0]
  4. Computational Science – ICCS 2020

    Projective Integration for Moment Models of the BGK Equation

    Koellermeier, Julian | Samaey, Giovanni

    2020

    https://doi.org/10.1007/978-3-030-50433-5_25 [Citations: 1]
  5. Error estimators for adaptive simulation of rarefied gases using hyperbolic moment models

    Koellermeier, Julian

    31ST INTERNATIONAL SYMPOSIUM ON RAREFIED GAS DYNAMICS: RGD31, (2019), P.120004

    https://doi.org/10.1063/1.5119617 [Citations: 1]
  6. Machine learning moment closure models for the radiative transfer equation I: Directly learning a gradient based closure

    Huang, Juntao | Cheng, Yingda | Christlieb, Andrew J. | Roberts, Luke F.

    Journal of Computational Physics, Vol. 453 (2022), Iss. P.110941

    https://doi.org/10.1016/j.jcp.2022.110941 [Citations: 14]
  7. Two-Dimensional Simulation of Rarefied Gas Flows Using Quadrature-Based Moment Equations

    Koellermeier, Julian | Torrilhon, Manuel

    Multiscale Modeling & Simulation, Vol. 16 (2018), Iss. 2 P.1059

    https://doi.org/10.1137/17M1147548 [Citations: 11]
  8. Hermite spectral method for multi-species Boltzmann equation

    Li, Ruo | Lu, Yixiao | Wang, Yanli | Xu, Haoxuan

    Journal of Computational Physics, Vol. 471 (2022), Iss. P.111650

    https://doi.org/10.1016/j.jcp.2022.111650 [Citations: 3]
  9. Aggregation and disaggregation of active particles on the unit sphere with time-dependent frequencies

    Kim, Dohyun | Kim, Jeongho

    Discrete & Continuous Dynamical Systems - B, Vol. 27 (2022), Iss. 4 P.2247

    https://doi.org/10.3934/dcdsb.2021131 [Citations: 0]
  10. Hyperbolic Problems: Theory, Numerics, Applications. Volume II

    Numerical Approximation of a Simplified Kinetic Model for a Sedimenting Suspension of Rod-Like Particles Using Hyperbolic Systems of Moment Equations

    Dahm, Sina | Helzel, Christiane

    2024

    https://doi.org/10.1007/978-3-031-55264-9_29 [Citations: 0]
  11. Moment models for neutral particles in high-collisional regimes for plasma edge simulations

    Lopez, Luis Fernando Cusicanqui | Koellermeier, Julian | Maes, Vince | Samaey, Giovanni

    2ND INTERNATIONAL CONFERENCE ON ADVANCED EARTH SCIENCE AND FOUNDATION ENGINEERING (ICASF 2023): Advanced Earth Science and Foundation Engineering, (2024), P.190002

    https://doi.org/10.1063/5.0187540 [Citations: 0]
  12. Projective integration schemes for hyperbolic moment equations

    Koellermeier, Julian | Samaey, Giovanni

    Kinetic & Related Models, Vol. 14 (2021), Iss. 2 P.353

    https://doi.org/10.3934/krm.2021008 [Citations: 7]
  13. Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps

    Koellermeier, Julian | Samaey, Giovanni

    The SMAI Journal of computational mathematics, Vol. 8 (2022), Iss. P.295

    https://doi.org/10.5802/smai-jcm.88 [Citations: 1]
  14. Accelerating the Convergence of the Moment Method for the Boltzmann Equation Using Filters

    Fan, Yuwei | Koellermeier, Julian

    Journal of Scientific Computing, Vol. 84 (2020), Iss. 1

    https://doi.org/10.1007/s10915-020-01251-8 [Citations: 16]
  15. Entropy bounds for the space–time discontinuous Galerkin finite element moment method applied to the BGK–Boltzmann equation

    Abdelmalik, M.R.A. | van der Woude, D.A.M. | van Brummelen, E.H.

    Computer Methods in Applied Mechanics and Engineering, Vol. 398 (2022), Iss. P.115162

    https://doi.org/10.1016/j.cma.2022.115162 [Citations: 1]
  16. Hierarchical micro-macro acceleration for moment models of kinetic equations

    Koellermeier, Julian | Vandecasteele, Hannes

    Journal of Computational Physics, Vol. 488 (2023), Iss. P.112194

    https://doi.org/10.1016/j.jcp.2023.112194 [Citations: 1]
  17. An Approximation for the Twenty-One-Moment Maximum-Entropy Model of Rarefied Gas Dynamics

    Giroux, Fabien | McDonald, James G.

    International Journal of Computational Fluid Dynamics, Vol. 35 (2021), Iss. 8 P.632

    https://doi.org/10.1080/10618562.2022.2047666 [Citations: 7]
  18. Theory, Numerics and Applications of Hyperbolic Problems II

    Simplified Hyperbolic Moment Equations

    Koellermeier, Julian | Torrilhon, Manuel

    2018

    https://doi.org/10.1007/978-3-319-91548-7_17 [Citations: 0]
  19. Hermite Spectral Method for Multi-Species Boltzmann Equation

    Li, Ruo | Lu, Yixiao | Wang, Yanli | Xu, Haoxuan

    SSRN Electronic Journal , Vol. (2022), Iss.

    https://doi.org/10.2139/ssrn.4062737 [Citations: 1]
  20. fenicsR13

    Theisen, Lambert | Torrilhon, Manuel

    ACM Transactions on Mathematical Software, Vol. 47 (2021), Iss. 2 P.1

    https://doi.org/10.1145/3442378 [Citations: 4]
  21. Spline moment models for the one-dimensional Boltzmann–Bhatnagar–Gross–Krook equation

    Koellermeier, Julian | Scholz, Ullika

    Physics of Fluids, Vol. 32 (2020), Iss. 10

    https://doi.org/10.1063/5.0020998 [Citations: 4]
  22. Entropy stable Hermite approximation of the linearised Boltzmann equation for inflow and outflow boundaries

    Sarna, Neeraj | Torrilhon, Manuel

    Journal of Computational Physics, Vol. 369 (2018), Iss. P.16

    https://doi.org/10.1016/j.jcp.2018.04.050 [Citations: 8]
  23. Hierarchical Boltzmann simulations and model error estimation

    Torrilhon, Manuel | Sarna, Neeraj

    Journal of Computational Physics, Vol. 342 (2017), Iss. P.66

    https://doi.org/10.1016/j.jcp.2017.04.041 [Citations: 21]
  24. Moment theories for a -dimensional dilute granular gas of Maxwell molecules

    Gupta, Vinay Kumar

    Journal of Fluid Mechanics, Vol. 888 (2020), Iss.

    https://doi.org/10.1017/jfm.2020.20 [Citations: 5]