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Journal of Scientific Computing

Erschienen in:

11.08.2018

Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations

verfasst von: M. Paul Laiu, Cory D. Hauck

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2019

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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.

Vorheriger Artikel A Posteriori Error Analysis of a Mixed-Primal Finite Element Method for the Boussinesq Problem with Temperature-Dependent Viscosity

Nächster Artikel Domain Decomposition Methods Using Dual Conversion for the Total Variation Minimization with Fidelity Term

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In this paper, the term “concentration” refers the integral of the kinetic distribution over the momentum/angular space. The concentration is a function of position and time only.

In general, \(n = (N+1)^2\); however, in reduced geometries, the components of \(\mathbf {m}\) are not all linearly independent. In such cases, n will be smaller.

Note that \(\mathbf {m}\) depends on N and could be denoted as \(\mathbf {m}_N\). We suppress this notation here for the sake of simplicity.

See, for example, [15, 42] for the detailed formulation of the FP\(_N\) equations.

Note that the results presented in this section only focus on the consistency properties of the limiters. A full convergence analysis for the P\(_N\) and FP\(_N\) equations with limiters is in the scope of future work.

Here we define the Sobolev spaces on \([-1,1]\) in (19) using weak derivatives and space interpolations as in [8, 41]. It is known that the Sobolev–Slobodeckij spaces used in [5] are equivalent to (19). On the other hand, we define the Sobolev spaces on \(\mathbb {S}^2\) in (20) via expansion coefficients as in [12, 20]. In [20, Section 8.1], it is shown that (20) is equivalent to a norm based on weak derivatives and space interpolations on \(\mathbb {S}^2\). We choose to use the form in (20) for simplicity.

Here, for \(q\in \mathbb {N}{\setminus }\{0\}\), \(C^q(\mathcal {S})\) is the space of functions with a continuous qth derivative on \(\mathcal {S}\). For \(q\ge 0\), \(C^q(\mathcal {S})\) is defined by norms given in [25, Eq. (4.3)–(4.4)]. See also [12, 36] for further details.

This is primarily due to the fact that, on \(\mathbb {S}^2\), more points are required for quadratures to achieve precision \(2N+1\). This means that there is generally no polynomial interpolant of \(\varphi \) in \(\mathbb {P}_N\), thus the second equality in (31) does not hold on \(\mathbb {S}^2\).

For general problems, it may not be possible to take advantage of symmetries.

It is reported in [25] that the computational time of the opt limiter can be reduced by about 30% with the tensor product quadrature replaced by the Lebedev quadrature [26]. However, the implementation of the opt limiter is still relatively expensive even with such improvement.

Note that the negative particle concentrations are colored in white.

Similar results are obtained for the \(L^1\) and \(L^\infty \) spatial errors.

The time step \(\varDelta t\) is also refined in such a way that the ratio \(\varDelta t/h\) stays fixed.

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)MathSciNetCrossRefMATH

Atkinson, K.: Numerical integration on the sphere. J. Aust. Math. Soc. Ser. B 23, 332–347 (1982)MathSciNetCrossRefMATH

Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Springer, Berlin (2012)CrossRefMATH

Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1976)CrossRefMATH

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 MathSciNetCrossRefMATH

Brunner, T.A.: Forms of approximate radiation transport. Technical Report SAND2002-1778, Sandia National Laboratories (2002)

Brunner, T.A., Holloway, J.P.: Two-dimensional time-dependent Riemann solvers for neutron transport. J. Comput. Phys. 210, 386–399 (2005)MathSciNetCrossRefMATH

Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38(157), 67–86 (1982)MathSciNetCrossRefMATH

Case, K., Zweifel, P.: Linear Transport Theory. Addison-Wesley, Reading, MA (1967)MATH

10.

Cercignani, C.: The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol. 67. Springer, New York (1988)CrossRefMATH

11.

Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106. Springer, New York (1994)CrossRefMATH

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.

Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, Volume 6: Evolution Problems II. Spinger, Berlin (2000)CrossRefMATH

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 86-0275

15.

Frank, M., Hauck, C., Küpper, K.: Convergence of filtered spherical harmonic equations for radiation transport. Commun. Math. Sci. 14(5), 1443–1465 (2016)MathSciNetCrossRefMATH

16.

Ganapol, B.D.: Homogeneous infinite media time-dependent analytic benchmarks for X-TM transport methods development. Technical report, Los Alamos National Laboratory (1999)

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)CrossRefMATH

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.

Gottlieb, D., Gottlieb, S., Hesthaven, J.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, New York (2007)MATH

20.

Guo, B.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)CrossRefMATH

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)MathSciNetCrossRefMATH

22.

Hauck, C.D., McClarren, R.G.: Positive \({P_N}\) closures. SIAM J. Sci. Comput. 32(5), 2603–2626 (2010)MathSciNetCrossRefMATH

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.

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)

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)MathSciNetCrossRefMATH

26.

Lebedev, V.: Quadratures on a sphere. Comput. Math. Math. Phys. 16, 10–24 (1976)MathSciNetCrossRefMATH

27.

LeVeque, R.: Finite difference methods for ordinary and partial differential equations. Soc. Ind. Appl. Math. (2007). https://doi.org/10.1137/1.9780898717839 MATH

28.

Lewis, E.E., Miller, W.F.J.: Computational Methods in Neutron Transport. Wiley, New York (1984)

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 CrossRef

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)MathSciNetCrossRefMATH

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 MathSciNetCrossRefMATH

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.

Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer, New York (1990)CrossRefMATH

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.

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)MathSciNetCrossRefMATH

36.

Nezza, E.D., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefMATH

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 MathSciNetCrossRefMATH

38.

Perthame, B.: Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27(6), 1405–1421 (1990)MathSciNetCrossRefMATH

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 MathSciNetCrossRefMATH

40.

Pomraning, G.C.: Radiation Hydrodynamics. Pergamon Press, New York (1973)

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 CrossRefMATH

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.

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.

Walters, W.: Use of the Chebyshev–Legendre quadrature set in discrete-ordinate codes. Technical report, LA-UR-87-3621, Los Alamos National Laboratory (1987)

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.

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 MathSciNetCrossRefMATH

Titel: Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations
verfasst von: M. Paul Laiu
Cory D. Hauck
Publikationsdatum: 11.08.2018
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2/2019
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0790-y

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