Advertisement

Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations

  • M. Paul Laiu
  • Cory D. Hauck
Article
  • 18 Downloads

Abstract

We analyze the properties and compare the performance of several positivity limiters for spectral approximations with respect to the angular variable of linear transport equations. It is well-known that spectral methods suffer from the occurrence of (unphysical) negative spatial particle concentrations due to the fact that the underlying polynomial approximations are not always positive at the kinetic level. Positivity limiters address this defect by enforcing positivity of the polynomial approximation on a finite set of preselected points. With a proper PDE solver, they ensure positivity of the particle concentration at each step in a time integration scheme. We review several known positivity limiters proposed in other contexts and also introduce a modification for one of them. We give error estimates for the consistency of the positive approximations produced by these limiters and compare the theoretical estimates to numerical results. We then solve two benchmark problems with these limiters, make qualitative and quantitative observations about the solutions, and then compare the efficiency of the different limiters.

Keywords

Kinetic equation Spectral methods Positivity-preserving limiters Filters 

Mathematics Subject Classification

35L02 41A10 41A25 41A36 42B37 65M70 82C70 82D75 

References

  1. 1.
    Alldredge, G.W., Hauck, C.D., Tits, A.L.: High-order entropy-based closures for linear transport in slab geometry II: a computational study of the optimization problem. SIAM J. Sci. Comput. 34(4), B361–B391 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Atkinson, K.: Numerical integration on the sphere. J. Aust. Math. Soc. Ser. B 23, 332–347 (1982)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Springer, Berlin (2012)CrossRefMATHGoogle Scholar
  4. 4.
    Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1976)CrossRefMATHGoogle Scholar
  5. 5.
    Bernardi, C., Maday, Y.: Polynomial interpolation results in Sobolev spaces. J. Comput. Appl. Math. 43(1), 53–80 (1992).  https://doi.org/10.1016/0377-0427(92)90259-Z MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brunner, T.A.: Forms of approximate radiation transport. Technical Report SAND2002-1778, Sandia National Laboratories (2002)Google Scholar
  7. 7.
    Brunner, T.A., Holloway, J.P.: Two-dimensional time-dependent Riemann solvers for neutron transport. J. Comput. Phys. 210, 386–399 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38(157), 67–86 (1982)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Case, K., Zweifel, P.: Linear Transport Theory. Addison-Wesley, Reading, MA (1967)MATHGoogle Scholar
  10. 10.
    Cercignani, C.: The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol. 67. Springer, New York (1988)CrossRefMATHGoogle Scholar
  11. 11.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106. Springer, New York (1994)CrossRefMATHGoogle Scholar
  12. 12.
    Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, New York (2013). https://books.google.com/books?id=8SA_AAAAQBAJ. Accesses 2013
  13. 13.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, Volume 6: Evolution Problems II. Spinger, Berlin (2000)CrossRefMATHGoogle Scholar
  14. 14.
    Deshpande, S.M.: Kinetic theory based new upwind methods for inviscid compressible flows. In: American Institute of Aeronautics and Astronautics, New York (1986). Paper 86-0275Google Scholar
  15. 15.
    Frank, M., Hauck, C., Küpper, K.: Convergence of filtered spherical harmonic equations for radiation transport. Commun. Math. Sci. 14(5), 1443–1465 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ganapol, B.D.: Homogeneous infinite media time-dependent analytic benchmarks for X-TM transport methods development. Technical report, Los Alamos National Laboratory (1999)Google Scholar
  17. 17.
    Garrett, C.K., Hauck, C.D.: A comparison of moment closures for linear kinetic transport equations: the line source benchmark. Transp. Theory Stat. Phys. 42, 203–235 (2013)CrossRefMATHGoogle Scholar
  18. 18.
    Garrett, P.: Harmonic analysis on spheres, II (2011). http://www.math.umn.edu/~garrett/m/mfms/notes_c/spheres_II.pdf. Accesses 2011
  19. 19.
    Gottlieb, D., Gottlieb, S., Hesthaven, J.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, New York (2007)MATHGoogle Scholar
  20. 20.
    Guo, B.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)CrossRefMATHGoogle Scholar
  21. 21.
    Harten, A., Lax, P.D., Leer, V.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25, 35–61 (1983)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hauck, C.D., McClarren, R.G.: Positive \({P_N}\) closures. SIAM J. Sci. Comput. 32(5), 2603–2626 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Laboure, V.M., McClarren, R.G., Hauck, C.D.: Implicit filtered P\(_N\) for high-energy density thermal radiation transport using discontinuous Galerkin finite elements. J. Comput. Phys. 321, 624–643 (2016).  https://doi.org/10.1016/j.jcp.2016.05.046. http://www.sciencedirect.com/science/article/pii/S0021999116301917
  24. 24.
    Laiu, M.P.: Positive filtered P\(_n\) method for linear transport equations and the associated optimization algorithm. Ph.D. thesis, University of Maryland, College Park (2016)Google Scholar
  25. 25.
    Laiu, M.P., Hauck, C.D., McClarren, R.G., O’Leary, D.P., Tits, A.L.: Positive filtered P\(_{N}\) moment closures for linear kinetic equations. SIAM J. Numer. Anal. 54(6), 3214–3238 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Lebedev, V.: Quadratures on a sphere. Comput. Math. Math. Phys. 16, 10–24 (1976)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    LeVeque, R.: Finite difference methods for ordinary and partial differential equations. Soc. Ind. Appl. Math. (2007).  https://doi.org/10.1137/1.9780898717839 MATHGoogle Scholar
  28. 28.
    Lewis, E.E., Miller, W.F.J.: Computational Methods in Neutron Transport. Wiley, New York (1984)Google Scholar
  29. 29.
    Light, D., Durran, D.: Preserving nonnegativity in discontinuous Galerkin approximations to scalar transport via truncation and mass aware rescaling (TMAR). Mon. Weather Rev. 144(12), 4771–4786 (2016).  https://doi.org/10.1175/MWR-D-16-0220.1 CrossRefGoogle Scholar
  30. 30.
    Liu, X.D., Osher, S.: Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I. SIAM J. Numer. Anal. 33(2), 760–779 (1996)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Liu, Y., Cheng, Y., Shu, C.W.: A simple bound-preserving sweeping technique for conservative numerical approximations. J. Sci. Comput. 73(2), 1028–1071 (2017).  https://doi.org/10.1007/s10915-017-0395-x MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Loubère, R., Staley, M., Wendroff, B.: The repair paradigm: new algorithms and applications to compressible flow. J. Comput. Phys. 211(2), 385–404 (2006).  https://doi.org/10.1016/j.jcp.2005.05.010. www.sciencedirect.com/science/article/pii/S0021999105002706
  33. 33.
    Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer, New York (1990)CrossRefMATHGoogle Scholar
  34. 34.
    McClarren, R.G., Hauck, C.D.: Robust and accurate filtered spherical harmonics expansions for radiative transfer. J. Comput. Phys. 229(16), 5597–5614 (2010).  https://doi.org/10.1016/j.jcp.2010.03.043. http://www.sciencedirect.com/science/article/pii/S0021999110001622
  35. 35.
    McClarren, R.G., Holloway, J.P., Brunner, T.A.: On solutions to the \({P_N}\) equations for thermal radiative transfer. J. Comput. Phys. 227(5), 2864–2885 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Nezza, E.D., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Olson, G.L.: Second-order time evolution of \({P_N}\) equations for radiation transport. J. Comput. Phys. 228(8), 3072–3083 (2009).  https://doi.org/10.1016/j.jcp.2009.01.012 MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Perthame, B.: Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27(6), 1405–1421 (1990)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Perthame, B.: Second-order Boltzmann schemes for compressible euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29(1), 1–19 (1992).  https://doi.org/10.1137/0729001 MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Pomraning, G.C.: Radiation Hydrodynamics. Pergamon Press, New York (1973)Google Scholar
  41. 41.
    Quarteroni, A.: Some results of Bernstein and Jackson type for polynomial approximation in \({L^p}\)-spaces. Jpn. J. Appl. Math. 1(1), 173–181 (1984).  https://doi.org/10.1007/BF03167866 CrossRefMATHGoogle Scholar
  42. 42.
    Radice, D., Abdikamalov, E., Rezzolla, L., Ott, C.D.: A new spherical harmonics scheme for multi-dimensional radiation transport I: static matter configurations. J. Comput. Phys. 242(0), 648–669 (2013).  https://doi.org/10.1016/j.jcp.2013.01.048. http://www.sciencedirect.com/science/article/pii/S0021999113001125
  43. 43.
    Shashkov, M., Wendroff, B.: The repair paradigm and application to conservation laws. J. Comput. Phys. 198(1), 265–277 (2004).  https://doi.org/10.1016/j.jcp.2004.01.014. www.sciencedirect.com/science/article/pii/S0021999104000270
  44. 44.
    Walters, W.: Use of the Chebyshev–Legendre quadrature set in discrete-ordinate codes. Technical report, LA-UR-87-3621, Los Alamos National Laboratory (1987)Google Scholar
  45. 45.
    Zhang, X., Shu, C.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229(9), 3091–3120 (2010).  https://doi.org/10.1016/j.jcp.2009.12.030. http://www.sciencedirect.com/science/article/pii/S0021999109007165
  46. 46.
    Zhang, X., Shu, C.W.: Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 467(2134), 2752–2776 (2011).  https://doi.org/10.1098/rspa.2011.0153 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply  2018

Authors and Affiliations

  1. 1.Computational and Applied Mathematics Group, Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  2. 2.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

Personalised recommendations