Skip to main content
SpringerLink
Log in

Adaptive Finite Element Simulation of the Time-dependent Simplified P N Equations

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

Abstract

The steady-state simplified P N approximation to the radiative transfer equation has been successfully applied to many problems involving radiation. Recently, time-dependent simplified P N equations have been derived by an asymptotic analysis similar to the asymptotic derivation of the steady-state SP N equations (Frank et al. in J. Comput. Phys. 226:2289–2305, 2007). In this paper, we present computational results for the time-dependent SP N equations in two dimensions, obtained by using an adaptive finite element approach. Several numerical comparisons with other existing models are shown.

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

Similar content being viewed by others

References

  1. Baudron, A.-M., Lautard, J.-J.: MINOS: a simplified pn solver for core calculation. Nucl. Sci. Eng. 155, 250–263 (2007)

    Google Scholar 

  2. Baudron, A.M., Geérin, P., Lautard, J.J.: Domain decomposition methods for core calculations using the MINOS solver. In: Joint International Topical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications (2007)

    Google Scholar 

  3. Bornemann, F.: An adaptive multilevel approach to parabolic equations. III. 2D error estimation and multilevel preconditioning. Comput. Sci. Eng. 4, 1–45 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Brantley, P.S., Larsen, E.W.: The simplified p3 approximation. Nucl. Sci. Eng. 134, 1 (2000)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Deuflhard, P., Bornemann, F.: Scientific Computing with Ordinary Differential Equations. Texts in Applied Mathematics, vol. 42. Springer, Berlin (2002)

    MATH  Google Scholar 

  7. Deuflhard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. Comput. Sci. Eng. 1, 3–35 (1989)

    MATH  Google Scholar 

  8. Erdmann, B., Lang, J., Roitzsch, R.: Kardos User’s Guide. Konrad–Zuse–Zentrum, Berlin (2002)

    Google Scholar 

  9. Frank, M., Klar, A., Larsen, E.W., Yasuda, S.: Time-dependent simplified pn approximation to the equations of radiative transfer. J. Comput. Phys. 226, 2289–2305 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gelbard, E.M.: Applications of spherical harmonics method to reactor problems. Tech. report WAPD-BT-20, Bettis Atomic Power Laboratory (1960)

  11. Gelbard, E.M.: Simplified spherical harmonics equations and their use in shielding problems. Tech. report WAPD-T-1182, Bettis Atomic Power Laboratory (1961)

  12. Gelbard, E.M.: Applications of the simplified spherical harmonics equations in spherical geometry. Tech. report WAPD-TM-294, Bettis Atomic Power Laboratory (1962)

  13. Gustafsson, K., Lundh, M., Söderlind, G.: A PI stepsize control for the numerical solution of ordinary differential equations. BIT Numer. Math. 28, 270–287 (1988)

    Article  MATH  Google Scholar 

  14. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Equations. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (1996)

    MATH  Google Scholar 

  15. Klar, A., Lang, J., Seaïd, M.: Adaptive solution of SP N -approximations to radiative heat transfer in glass. Int. J. Therm. Sci. 44, 1013–1023 (2005)

    Article  Google Scholar 

  16. Lang, J.: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications. Lecture Notes in Computational Science and Engineering, vol. 16. Springer, Berlin (2000)

    Google Scholar 

  17. Lang, J., Teleaga, D.: Towards a fully space-time adaptive FEM for magnetoquasistatics. IEEE Trans. Magn. 44, 1238–1241 (2008)

    Article  Google Scholar 

  18. Lang, J., Walter, A.: A finite element method adaptive in space and time for nonlinear reaction–diffusion systems. Comput. Sci. Eng. 4, 269–314 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Larsen, E.W., Morel, J.E., McGhee, J.M.: Asymptotic derivation of the multigroup p1 and simplified pn equations with anisotropic scattering. Nucl. Sci. Eng. 123, 328 (1996)

    Google Scholar 

  20. Larsen, E.W., Thömmes, G., Klar, A.: New frequency–averaged approximations to the equations of radiative heat transfer. SIAM J. Appl. Math. 64, 565–582 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Larsen, E.W., Thömmes, G., Klar, A., Seaïd, M., Götz, T.: Simplified P n approximations to the equations of radiative heat transfer in glass. J. Comput. Phys. 183, 652–675 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marshak, R.E.: Note on the spherical harmonic method as applied to the Milne problem for a sphere. Phys. Rev. 71, 443–446 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  23. Morel, J.E., McGhee, J.M., Larsen, E.W.: A three-dimensional time-dependent unstructured tetrahedral-mesh SP N method. Nucl. Sci. Eng. 123, 319–327 (1996)

    Google Scholar 

  24. Su, B., Olson, G.L.: An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium. Ann. Nucl. Energy 24, 1035–1055 (1997)

    Article  Google Scholar 

  25. Tomasevic, D.I., Larsen, E.W.: The simplified P2 approximation. Nucl. Sci. Eng. 122, 309–325 (1996)

    Google Scholar 

  26. van der Vorst, H.A.: BI-CGSTAB a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics, Center for Computational Engineering Science, RWTH Aachen University, Schinkelstrasse 2, 52062, Aachen, Germany

    Martin Frank

  2. Mathematics, DFG Graduate School of Excellence Computational Engineering, DFG Excellence Cluster Smart Interfaces, TU Darmstadt, Dolivostrasse 15, 64293, Darmstadt, Germany

    Jens Lang

  3. Fraunhofer ITWM, Fraunhoferplatz 1, 67663, Kaiserslautern, Germany

    Matthias Schäfer

Authors
  1. Martin Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Lang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthias Schäfer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Frank.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frank, M., Lang, J. & Schäfer, M. Adaptive Finite Element Simulation of the Time-dependent Simplified P N Equations. J Sci Comput 49, 332–350 (2011). https://doi.org/10.1007/s10915-011-9466-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-011-9466-6

Keywords

Search

Navigation