Skip to main content
SpringerLink
Log in

Implicit Discontinuous Galerkin Method for the Boltzmann Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

An implicit high-order discontinuous Galerkin (DG) method is developed to find the steady-state solution of rarefied gas flow described by the Boltzmann equation with full collision operator. In the physical space, the velocity distribution function is approximated by the piecewise polynomial of degree up to 4, while in the velocity space the fast spectral method is incorporated into the DG discretization to evaluate the collision operator. A specific polynomial approximation for the collision operator is proposed to reduce the computational cost of the fast spectral method by K times, where for two-dimensional problem K is 15 when the DG with 4th-order polynomial is used on triangular mesh. Based on the first-order upwind scheme, a sweeping technique is employed to solve the local linear equations resulting from the DG discretization sequentially over spatial elements. This technique can preserve stability of the scheme and requires no nonlinear limiter in solving hypersonic rarefied gas flows when the shock wave structure is fully resolved by fine spatial grid. Moreover, without assembling large sparse linear system, the computational cost in terms of memory consumption and CPU time is significantly reduced. Five different one/two-dimensional tests including low-speed microscale flows and hypersonic rarefied gas flows are used to assess the accuracy and efficiency of proposed approach. Our results show that, DG scheme of different order of approximating polynomial requires the same number of iterative steps to obtain the steady-state solution with the same order of accuracy; and the higher order the scheme is, the fewer spatial elements are needed, thus leading to less CPU time. Besides, our method can be faster than the finite difference solver by about one order of magnitude. The produced solutions can be used as benchmark data for assessing the accuracy of other gas kinetic solvers for the Boltzmann equation and gas kinetic models that simplify the Boltzmann collision operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Alekseenko, A., Nguyen, T., Wood, A.: A deterministic-stochastic method for computing the Boltzmann collision integral in \({\cal{O}}(mn)\) operations. Kinetic Relat. Models 11(5), 1211–1234 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Alsmeyer, H.: Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech. 74(3), 497–513 (1976)

    Google Scholar 

  3. Aristov, V.V.: Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Springer, Dordrecht (2001)

    MATH  Google Scholar 

  4. Baker, L.L., Hadjiconstantinou, N.G.: Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: a discontinuous Galerkin formulation. Int. J. Numer. Methods Fluids 58(4), 381–402 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)

    MATH  Google Scholar 

  6. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation. Clarendon, Oxford (1994)

    Google Scholar 

  7. Bobylev, A.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Math. Phys. Rev. 7, 111–233 (1988)

    MathSciNet  MATH  Google Scholar 

  8. Bobylev, A., Palczewski, A., Schneider, J.: On approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. 320, 639–644 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Bobylev, A., Rjasanow, R.: Difference scheme for the Boltzmann equation based on fast Fourier transformation. Eur. J. Mech. B Fluids 16(2), 293–306 (1997)

    MATH  Google Scholar 

  10. Bobylev, A., Rjasanow, S.: Fast deterministic method of solving the Boltzmann equation for hard spheres. Eur. J. Mech. B Fluids 18(5), 869–887 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Buet, C.: A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transp. Theory Stat. Phys. 25(1), 33–60 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Chai, J.C., Lee, H.S., Patankar, S.V.: Ray effect and false scattering in the discrete ordinates method. Numer. Heat Transf. Part B Fund. 24(4), 373–389 (1993)

    Google Scholar 

  13. Chapman, S., Cowling, T.: The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge University Press, New York (1970)

    MATH  Google Scholar 

  14. Chen, S., Zhang, C., Zhu, L., Guo, Z.: A unified implicit scheme for kinetic model equations. Part I. Memory reduction technique. Sci. Bull. 62(2), 119–129 (2017)

    Google Scholar 

  15. Cockburn, B., Shu, C.W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Cockburn, B., Shu, C.W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Coelho, P.: The role of ray effects and false scattering on the accuracy of the standard and modified discrete ordinates methods. J. Quant. Spectrosc. Radiat. Transf. 73(2), 231–238 (2002)

    Google Scholar 

  18. Crivellini, A., Bassi, F.: An implicit matrix-free discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations. Comput. Fluids 50(1), 81–93 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Crouseilles, N., Mehrenberger, M., Sonnendrücker, E.: Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229(6), 1927–1953 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Crouseilles, N., Respaud, T., Sonnendrücker, E.: A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Commun. 180(10), 1730–1745 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Dimarco, G., Loubere, R.: Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation. J. Comput. Phys. 255, 680–698 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Dimarco, G., Loubere, R.: Towards an ultra efficient kinetic scheme. Part II: the high order case. J. Comput. Phys. 255, 699–719 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Dimarco, G., Loubère, R., Narski, J., Rey, T.: An efficient numerical method for solving the Boltzmann equation in multidimensions. J. Comput. Phys. 353, 46–81 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Dimarco, G., Pareschi, L.: Numerical methods for kinetic equations. Acta Numer. 23, 369–520 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Filbet, F., Mouhot, C., Pareschi, L.: Solving the Boltzmann equation in \(n\log ^2n\). SIAM J. Sci. Comput. 28(3), 1029–1053 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Filbet, F., Russo, G.: High order numerical methods for the space non-homogeneous Boltzmann equation. J. Comput. Phys. 186(2), 457–480 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Gobbert, M., Webster, S., Cale, T.: A Galerkin method for the simulation of the transient 2-D/2-D and 3-D/3-D linear Boltzmann equation. J. Sci. Comput. 30(2), 237–273 (2007)

    MathSciNet  MATH  Google Scholar 

  28. Goldstein, D., Sturtevant, B., Broadwell, J.E.: Investigations of the motion of discrete-velocity gases. Prog. Astron. Aeron. 117, 100–117 (1989)

    Google Scholar 

  29. Guo, Z., Wang, R., Xu, K.: Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case. Phys. Rev. E 91(3), 033313 (2015)

    Google Scholar 

  30. Guo, Z., Xu, K., Wang, R.: Discrete unified gas kinetic scheme for all Knudsen number flows: low-speed isothermal case. Phys. Rev. E 88, 033305 (2013)

    Google Scholar 

  31. Güçlü, Y., Hitchon, W.: A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows. J. Comput. Phys. 231(8), 3289–3316 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Holway, L.H.: New statistical models for kinetic theory: methods of construction. Phys. Fluids 9(9), 1658–1673 (1966)

    Google Scholar 

  33. Huang, A.B., Giddens, D.P.: The discrete ordinate method for the linearized boundary value problems in kinetic theory of gases. In: Brundin C.L. (ed.) Rarefied Gas Dynamics, vol. 1, p. 481 (1967)

  34. Ibragimov, I., Rjasanow, S.: Numerical solution of the Boltzmann equation on the uniform grid. Computing 69(2), 163–186 (2002)

    MathSciNet  MATH  Google Scholar 

  35. Jaiswal, S., Alexeenko, A.A., Hu, J.: A discontinuous Galerkin fast spectral method for the full Boltzmann equation with general collision kernels. J. Comput. Phys. 378, 178–208 (2019)

    MathSciNet  MATH  Google Scholar 

  36. John, B., Gu, X.J., Emerson, D.R.: Investigation of heat and mass transfer in a lid-driven cavity under nonequilibrium flow conditions. Numer. Heat Transf. Part B Fund. 58(5), 287–303 (2010)

    Google Scholar 

  37. Kitzler, G., Schöberl, J.: A high order space-momentum discontinuous Galerkin method for the Boltzmann equation. Comput. Math. Appl. 70(7), 1539–1554 (2015)

    MathSciNet  Google Scholar 

  38. Kolobov, V., Arslanbekov, R., Aristov, V., Frolova, A., Zabelok, S.: Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement. J. Comput. Phys. 223(2), 589–608 (2007)

    MATH  Google Scholar 

  39. Kosuge, S., Aoki, K., Takata, S.: Shock-wave structure for a binary gas mixture: finite-difference analysis of the Boltzmann equation for hard-sphere molecules. Eur. J. Mech. B Fluids 20(1), 87–126 (2001)

    MATH  Google Scholar 

  40. Kubatko, E.J., Dawson, C., Westerink, J.J.: Time step restrictions for Runge–Kutta discontinuous Galerkin methods on triangular grids. J. Comput. Phys. 227(23), 9697–9710 (2008)

    MathSciNet  MATH  Google Scholar 

  41. Lewis, E., Miller, W.: Computational Methods of Neutron Transport. Wiley, New York (1984)

    MATH  Google Scholar 

  42. Liu, C., Xu, K., Sun, Q., Cai, Q.: A unified gas-kinetic scheme for continuum and rarefied flows IV: full Boltzmann and model equations. J. Comput. Phys. 314, 305–340 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Mieussens, L.: Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J. Comput. Phys. 162(2), 429–466 (2000)

    MathSciNet  MATH  Google Scholar 

  44. Morris, A.B., Varghese, P.L., Goldstein, D.B.: Improvement of a discrete velocity Boltzmann equation solver with arbitrary post-collision velocities. AIP Conf. Proc. 1084(1), 458–463 (2008)

    Google Scholar 

  45. Mouhot, C., Pareschi, L.: Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75(256), 1833–1852 (2006)

    MathSciNet  MATH  Google Scholar 

  46. Murphy, S.: Methods for solving discontinuous-Galerkin finite element equations with application to neutron transport. Ph.D. Thesis, Institute National Polytechnique de Toulouse (2015)

  47. Nabeth, J., Chigullapalli, S., Alexeenko, A.A.: Quantifying the Knudsen force on heated microbeams: a compact model and direct comparison with measurements. Phys. Rev. E 83, 066306 (2011)

    Google Scholar 

  48. Ohwada, T.: Structure of normal shock waves: direct numerical analysis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A Fluid Dyn. 5(1), 217–234 (1993)

    MathSciNet  MATH  Google Scholar 

  49. Ohwada, T.: Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite-difference analysis of the nonlinear Boltzmann equation for hard-sphere molecules. Phys. Fluids 8(8), 2153–2160 (1996)

    MATH  Google Scholar 

  50. Pareschi, L., Perthame, B.: A Fourier spectral method for homogeneous Boltzmann equations. Transp. Theory Stat. Phys. 25(3–5), 369–382 (1996)

    MathSciNet  MATH  Google Scholar 

  51. Pareschi, L., Russo, G.: Numerical solution of the Boltzmann equation I: spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37(4), 1217–1245 (2000)

    MathSciNet  MATH  Google Scholar 

  52. Passian, A., Warmack, R.J., Ferrell, T.L., Thundat, T.: Thermal transpiration at the microscale: a crookes cantilever. Phys. Rev. Lett. 90, 124503 (2003)

    Google Scholar 

  53. Radtke, G.A., Hadjiconstantinou, N.G., Wagner, W.: Low-noise Monte Carlo simulation of the variable hard sphere gas. Phys. Fluids 23(3), 030606 (2011)

    Google Scholar 

  54. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report, vol. 836 (1973)

  55. Shakhov, E.: Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 3(5), 95–96 (1968)

    MathSciNet  Google Scholar 

  56. Sharipov, F.: Modeling of transport phenomena in gases based on quantum scattering. Phys. A Stat. Mech. Appl. 508, 797–805 (2018)

    MathSciNet  Google Scholar 

  57. Sone, Y., Ohwada, T., Aoki, K.: Temperature jump and Knudsen layer in a rarefied gas over a plane wall: numerical analysis of the linearized Boltzmann equation for hard-sphere molecules. Phys. Fluids A Fluid Dyn. 1(2), 363–370 (1989)

    MATH  Google Scholar 

  58. Sone, Y., Yoshimoto, M.: Demonstration of a rarefied gas flow induced near the edge of a uniformly heated plate. Phys. Fluids 9(11), 3530–3534 (1997)

    Google Scholar 

  59. Strongrich, A., Alexeenko, A.: Microstructure actuation and gas sensing by the Knudsen thermal force. Appl. Phys. Lett. 107(19), 193508 (2015)

    Google Scholar 

  60. Su, W., Alexeenko, A.A., Cai, G.: A parallel Runge–Kutta discontinuous Galerkin solver for rarefied gas flows based on 2D Boltzmann kinetic equations. Comput. Fluids 109, 123–136 (2015)

    MathSciNet  MATH  Google Scholar 

  61. Su, W., Lindsay, S., Liu, H., Wu, L.: Comparative study of the discrete velocity and lattice Boltzmann methods for rarefied gas flows through irregular channels. Phys. Rev. E 96, 023309 (2017)

    Google Scholar 

  62. Tang, Z., He, B., Cai, G.: Investigation on a coupled Navier–Stokes direct simulation Monte Carlo method for the simulation of plume flowfield of a conical nozzle. Int. J. Numer. Methods Fluids 76(2), 95–108 (2014)

    MathSciNet  Google Scholar 

  63. Tcheremissine, F.: Conservative evaluation of Boltzmann collision integral in discrete ordinates approximation. Comput. Math. Appl. 35(1), 215–221 (1998)

    MathSciNet  MATH  Google Scholar 

  64. Tcheremissine, F.G.: Solution to the Boltzmann kinetic equation for high-speed flows. Comput. Math. Math. Phys. 46(2), 315–329 (2006)

    MathSciNet  Google Scholar 

  65. Titarev, V.A.: Efficient deterministic modelling of three-dimensional rarefied gas flows. Commun. Comput. Phys. 12(1), 162–192 (2012)

    MathSciNet  MATH  Google Scholar 

  66. Valentini, P., Schwartzentruber, T.E.: Large-scale molecular dynamics simulations of normal shock waves in dilute argon. Phys. Fluids 21(6), 066101 (2009)

    MATH  Google Scholar 

  67. Wagner, W.: Approximation of the Boltzmann equation by discrete velocity models. J. Stat. Phys. 78(5), 1555–1570 (1995)

    MathSciNet  MATH  Google Scholar 

  68. Watchararuangwita, C., Grigorievb, Y.N., Meleshkoa, S.V.: A deterministic spectral method for solving the Boltzmann equation for one-dimensional flows. Sci. Asia 35(1), 70–79 (2009)

    Google Scholar 

  69. Wu, L., Liu, H., Zhang, Y., Reese, J.M.: Influence of intermolecular potentials on rarefied gas flows: fast spectral solutions of the Boltzmann equation. Phys. Fluids 27(8), 082002 (2015)

    Google Scholar 

  70. Wu, L., Reese, J.M., Zhang, Y.: Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows. J. Fluid Mech. 746, 53–84 (2014)

    MathSciNet  MATH  Google Scholar 

  71. Wu, L., Struchtrup, H.: Assessment and development of the gas kinetic boundary condition for the Boltzmann equation. J. Fluid Mech. 823, 511–537 (2017)

    MathSciNet  MATH  Google Scholar 

  72. Wu, L., White, C., Scanlon, T.J., Reese, J.M., Zhang, Y.: Deterministic numerical solutions of the Boltzmann equation using the fast spectral method. J. Comput. Phys. 250, 27–52 (2013)

    MathSciNet  MATH  Google Scholar 

  73. Wu, L., Zhang, J., Liu, H., Zhang, Y., Reese, J.M.: A fast iterative scheme for the linearized Boltzmann equation. J. Comput. Phys. 338, 431–451 (2017)

    MathSciNet  MATH  Google Scholar 

  74. Wu, L., Zhang, J., Reese, J.M., Zhang, Y.: A fast spectral method for the Boltzmann equation for monatomic gas mixtures. J. Comput. Phys. 298, 602–621 (2015)

    MathSciNet  MATH  Google Scholar 

  75. Xu, K., Huang, J.C.: A unified gas-kinetic scheme for continuum and rarefied flows. J. Comput. Phys. 229(20), 7747–7764 (2010)

    MathSciNet  MATH  Google Scholar 

  76. Yang, J., Huang, J.: Rarefied flow computations using nonlinear model Boltzmann equations. J. Comput. Phys. 120(2), 323–339 (1995)

    MATH  Google Scholar 

  77. Yen, S.: Temperature overshoot in shock waves. Phys. Fluids 9(7), 1417–1418 (1966)

    Google Scholar 

  78. Zhu, L., Guo, Z.: Application of discrete unified gas kinetic scheme to thermally induced nonequilibrium flows. Comput. Fluids. 193, 103613 (2019)

    MathSciNet  MATH  Google Scholar 

  79. Zhu, T., Ye, W.: Origin of Knudsen forces on heated microbeams. Phys. Rev. E 82, 036308 (2010)

    Google Scholar 

  80. Zhu, Y., Zhong, C., Xu, K.: Implicit unified gas-kinetic scheme for steady state solutions in all flow regimes. J. Comput. Phys. 315, 16–38 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is founded by the Engineering and Physical Sciences Research Council (EPSRC) in the UK under Grant EP/R041938/1.

Author information

Authors and Affiliations

  1. James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow, G1 1XJ, UK

    Wei Su, Peng Wang & Yonghao Zhang

  2. Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, 518055, China

    Lei Wu

Authors
  1. Wei Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yonghao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lei Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Wu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Here, we present details of the DG formulation for the Boltzmann equation. The linear systems (25) to determine the solution of \(f^{j'}\) on spatial element \(\varDelta _i\) are recalled here:

$$\begin{aligned} {\mathbf {A}}^{i,j'}{\mathbf {F}}^{j'}_i+{\mathbf {B}}^{\text {ext},j'} ={\mathbf {S}}^{i,j'}, \end{aligned}$$
(A.37)

for \(i=1,\dots ,M_\text {el},\ j'=1,\dots ,M\).

We denote \(F^{j'}_r=F_r({\varvec{v}}^{j'})\), \(\varLambda ^{j'}_p=\varLambda _p({\varvec{v}}^{j'})\) and \(\varXi ^{j'}_{p,r}=\varXi _{p,r}({\varvec{v}}^{j'})\) as values of the corresponding variables at each discrete velocity point, and \({\mathbf {F}}^{j'}_i=[F^{j'}_1,\dots ,F^{j'}_r,\dots ]^\mathrm {T}\) is the vector of degrees of freedom of \(f^{j'}\) on \(\varDelta _i\). For the ITR-LOC scheme, the coefficient matrices are:

$$\begin{aligned}&{\mathbf {A}}^{i,j'}_{sr}=\frac{1}{2}\left( {\varvec{v}}^{j'}\cdot {\varvec{n}}+|{\varvec{v}}^{j'}\cdot {\varvec{n}}|\right) \langle \varphi _s,\varphi _r\rangle _{\partial \varDelta _i}-\left( {\varvec{v}}^{j'}\cdot \nabla \varphi _s,\varphi _r\right) _{\varDelta _i}+\sum ^{K}_{p=1} \left( \varphi _s,\varphi _p\varphi _r\right) _{\varDelta _i}\varLambda ^{j'}_p, \nonumber \\\end{aligned}$$
(A.38)
$$\begin{aligned}&{\mathbf {B}}^{\text {ext},j'}_{s}={\left\{ \begin{array}{ll} \frac{1}{2}\left( {\varvec{v}}^{j'}\cdot {\varvec{n}}-|{\varvec{v}}^{j'}\cdot {\varvec{n}}|\right) \sum ^{K}_{r=1}\langle \varphi _s,\varphi ^{\text {ext}}_r\rangle _{\partial \varDelta _i} F^{j'}_{r,\text {ext}}, \quad \partial \varDelta _i\not \subset \partial \varDelta \\ \frac{1}{2}\left( {\varvec{v}}^{j'}\cdot {\varvec{n}}-|{\varvec{v}}^{j'}\cdot {\varvec{n}}|\right) \langle \varphi _s,b^{j'}\rangle _{\partial \varDelta _i}, \quad \partial \varDelta _i\subset \partial \varDelta \end{array}\right. }, \end{aligned}$$
(A.39)
$$\begin{aligned}&{\mathbf {S}}_s=\sum ^{K}_{p=1}\sum ^{K}_{r=1}\left( \varphi _s,\varphi _p\varphi _r \right) _{\varDelta _i}\varXi ^{j'}_{p,r}, \end{aligned}$$
(A.40)

where \(\varphi ^{\text {ext}}_r\) denotes the supporting polynomials on the neighboring element, from which \(f_{\text {ext}}\) is obtained. For the ITR-MEAN scheme, the coefficient matrices become:

$$\begin{aligned}&{\mathbf {A}}^{i,j'}_{sr}=\frac{1}{2}\left( {\varvec{v}}^{j'}\cdot {\varvec{n}}+|{\varvec{v}}^{j'}\cdot {\varvec{n}}|\right) \langle \varphi _s,\varphi _r\rangle _{\partial \varDelta _i}-\left( {\varvec{v}}^{j'}\cdot \nabla \varphi _s,\varphi _r\right) _{\varDelta _i}+\left( \varphi _s, \varphi _r\right) _{\varDelta _i}{\bar{\nu }}, \nonumber \\\end{aligned}$$
(A.41)
$$\begin{aligned}&{\mathbf {B}}^{\text {ext},j'}_{s}={\left\{ \begin{array}{ll} \frac{1}{2}\left( {\varvec{v}}^{j'}\cdot {\varvec{n}}-|{\varvec{v}}^{j'}\cdot {\varvec{n}}|\right) \sum ^{K}_{r=1} \langle \varphi _s,\varphi ^{\text {ext}}_r\rangle _{\partial \varDelta _i} F^{j'}_{r,\text {ext}},\quad \partial \varDelta _i\not \subset \partial \varDelta \\ \frac{1}{2}\left( {\varvec{v}}^{j'}\cdot {\varvec{n}}-|{\varvec{v}}^{j'}\cdot {\varvec{n}}|\right) \langle \varphi _s,b^{j'}\rangle _{\partial \varDelta _i}, \quad \partial \varDelta _i\subset \partial \varDelta \end{array}\right. }, \end{aligned}$$
(A.42)
$$\begin{aligned}&{\mathbf {S}}_s=\sum ^{K}_{p=1}\sum ^{K}_{r=1}\left( \varphi _s,\varphi _p\varphi _r \right) _{\varDelta _i}\left( \varXi ^{j'}_{p,r}-\varLambda ^{j'}_pF^{j'}_r\right) +\sum ^{K}_{r=1}\left( \varphi _s,\varphi _r\right) _{\varDelta _i}{\bar{\nu }}F^{j'}_r. \end{aligned}$$
(A.43)

In this paper, nodal shape functions are chosen as the approximating polynomials. Integrals of the shape functions such as \(\left( \varphi _s,\varphi _r\right) \), \(\left( \nabla \varphi _s,\varphi _r\right) \), \(\left( \varphi _s,\varphi _p\varphi _r\right) \) and \(\langle \varphi _s,\varphi _r\rangle \) can be obtained analytically. To evaluate \(\langle \varphi _s,b^{j'}\rangle \), the Gaussian rule is applied.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Su, W., Wang, P., Zhang, Y. et al. Implicit Discontinuous Galerkin Method for the Boltzmann Equation. J Sci Comput 82, 39 (2020). https://doi.org/10.1007/s10915-020-01139-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01139-7

Keywords

Search

Navigation