The inner-element subgrid scale finite element method for the Boltzmann transport equation

Nuclear Science and Engineering

Volumn 164, Issue 2, 2010, Pages 105-121

  Buchan, Andrew G ^a   Candy, Adam S ^a   Merton, Simon R ^b   Pain, Christopher C ^a   Hadi, Justin I ^a   Eaton, Matthew D ^a   Goddard, Anthony J H ^a   Smedley Stevenson, Richard P ^a   Pearce, Gregory J ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b AWE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

BOLTZMANN EQUATION; GALERKIN METHODS; NUMERICAL METHODS; PERIODIC STRUCTURES; RADIATIVE TRANSFER;

BOLTZMANN TRANSPORT EQUATION; DISCONTINUOUS GALERKIN METHODS; FINITE-ELEMENT REPRESENTATION; GALERKIN FINITE ELEMENTS; LINEAR FINITE ELEMENTS; PETROV-GALERKIN METHODS; SPATIAL DISCRETIZATIONS; UNPHYSICAL OSCILLATIONS;

FINITE ELEMENT METHOD;

EID: 77649142634     PISSN: 00295639     EISSN: None     Source Type: Journal    
DOI: 10.13182/NSE08-82     Document Type: Article

References (29)

1. T. J. R. HUGHES, G. ENGEL, L. MAZZEI, and M. G. LARSON, "A Comparison of Discontinuous and Continuous Galerkin Methods Based on Error Estimates, Conservation, Robustness and Efficiency", Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, pp. 135-146, B. COCKBURN, G. E. KARNIADAKIS, and C.-W. SHU, Eds., Springer, Berlin (2000).
2. W. H. REED and T. R. HILL, "Triangular Mesh Methods for the Neutron Transport Equation", LA-UR-73-479, Los Alamos Scientific Laboratory (1973).
3. M. L. ADAMS, "Discontinuous Finite Element Transport Solutions in Thick Diffusive Problems", Nucl. Sci. Eng., 137, 298 (2001).
4. E. W. LARSEN, J. E. MOREL, and W. F. MILLER, Jr., "Asymptotic Solutions of Numerical Transport Problems in Optically Thick, Diffusive Regimes", J. Comput. Phys., 69, 283 (1986).
5. A. N. BROOKS and T. J. R. HUGHES, "Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations", Comput. Methods Appl. Mech. Eng., 32, 1-3, 199 (Sep. 1982).
6. C. C. PAIN, M. D. EATON, R. P. SMEDLEYSTEVENSON, A. J. H. GODDARD, M. D. PIGGOTT, and C. R. E. DE OLIVEIRA, "Streamline Upwind Petrov-Galerkin Methods for the Steady-State Boltzmann Transport Equation", Comput. Methods Appl. Mech. Eng., 195, 4448 (2006).
7. T. J. R. HUGHES, L. P. FRANCA, and G. M. HULBERT, "A New Finite Element Formulation for Computational Fluid Dynamics: VIII. The Galerkin/Least-Squares Method for Advective-Diffusive Equations", Comput. Methods Appl. Mech. Eng., 73, 173 (1989).
8. C. C. PAIN, M. D. EATON, R. P. SMEDLEYSTEVENSON, A. J. H. GODDARD, M. D. PIGGOTT, and C. R. E. DE OLIVEIRA, "Space-Time Streamline Upwind Petrov-Galerkin Methods for the Boltzmann Transport Equation", Comput. Methods Appl. Mech. Eng., 195, 4334 (2006).
9. C. BAIOCCHI, F. BREZZI, and L. P. FRANCA, "Virtual Bubbles and Galerkin-Least-Squares Type Methods (Ga. L. S.)", Comput. Methods Appl. Mech. Eng., 105, 125 (1993).
10. F. BREZZI, M. BRISTEAU, L. P. FRANCA, M. MALLET, and G. ROGÉ, "A Relationship Between Stabilized Finite Element Methods and the Galerkin Method with Bubble Functions", Comput. Methods Appl. Mech. Eng., 96, 117 (1992).
11. T. J. R. HUGHES, "Multiscale Phenomena: Green's Functions, the Dirichlet-to-Neumann Formulation, Subgrid Scale Models, Bubbles and the Origins of Stabilized Methods", Comput. Methods Appl. Mech. Eng., 127, 387 (1995).
12. L. P. FRANCA, A. NESLITURK, and M. STYNES, "On the Stability of Residual-Free Bubbles for Convection-Diffusion Problems and Their Approximation by a Two-Level Finite Element Method", Computer Methods Appl. Mech. Eng., 166, 1-2, 35 (Nov. 1998).
13. T. J. R. HUGHES, G. R. FEIJÓOM, L. MAZZEI, and J. B. QUINCY, "The Variational Multiscale Method-A Paradigm for Computational Mechanics", Comput. Methods Appl. Mech. Eng., 166, 3 (1998).
14. T. J. R. HUGHES, G. SCOVAZZI, P. B. BOCHEV, and A. BUFFA, "A Multiscale Discontinuous Galerkin Method with the Computational Structure of a Continuous Galerkin Method", Comput. Methods Appl. Mech. Eng., 195, 2761 (Apr. 2006).
15. R. J. LABEUR and G. N. WELLS, "A Galerkin Interface Stabilisation Method for the Advection-Diffusion and Incompressible Navier-Stokes Equations", Comput. Methods Appl. Mech. Eng., 196, 49-52, 4985 (2007).
16. G. SANGALLI, "A Discontinuous Residual-Free Bubble Method for Advection-Diffusion Problems", J. Eng. Math., 97, 779 (2004).
17. A. S. CANDY, C. C. PAIN, M. D. PIGGOTT, G. J. GORMAN, D. A. HAM, and R. P. SMEDLEY-STEVENSON, "A Subgrid Scale Stabilisation Method for the Advection-Reaction-Diffusion Equation" (in preparation).
18. E. E. LEWIS and W F. MILLER, Jr., Computational Methods of Neutron Transport, American Nuclear Society, La Grange Park, Illinois (1993).
19. A. G. BUCHAN, C. C. PAIN, M. D. EATON, R. P. SMEDLEY-STEVENSON, and A. J. H. GODDARD, "Linear and Quadratic Octahedral Wavelets on the Sphere for Angular Discretisations of the Boltzmann Transport Equation", Ann. Nucl. Energy, 32, 1224 (2005).
20. A. G. BUCHAN, C. C. PAIN, M. D. EATON, R. P. SMEDLEY-STEVENSON, and A. J. H. GODDARD, "Self-Adaptive Spherical Wavelets for Angular Discretizations of the Boltzmann Transport Equation", Nucl. Sci. Eng., 158, 244 (2008).
21. R. T. ACKROYD, Finite Element Methods for Particle Transport: Applications to Reactor and Radiation Physics, Research Studies Press, Taunton, Somerset, England (1997).
22. M. D. EATON, C. C. PAIN, C. R. E. DE OLIVEIRA, and A. J. H. GODDARD, "A High-Order Riemann Method for the Boltzmann Transport Equation", Proc. Nuclear Mathematical and Computational Sciences: A Century in Review, A Century Anew, Gatlinburg, Tennessee, April 6-11, 2003, American Nuclear Society (2003).
23. M. D. EATON, "A High-Resolution Riemann Method for Solving Radiation Transport Problems on Unstructured Meshes", PhD Thesis, University of London, Imperial College of Science, Technology and Medicine (2004).
24. Y. SAAD, Iterative Methods for Sparse Linear Systems (2000) ; available on the Internet at http://www-users.cs. umu.edu/~saad/books.html.
25. C. R. E. DE OLIVEIRA, C. C. PAIN, and M. D. EATON, "Hierarchical Angular Preconditioning for the Finite Element-Spherical Harmonics Radiation Transport Methods", Proc. PHYSOR 2000 ANS Int. Topi. Mtg. Advances in Reactor Physics and Mathematics and Computation into the Next Millennium, Pittsburgh, Pennsylvania, May 7-12, 2000, American Nuclear Society (2000).
26. S. R. MERTON, C. C. PAIN, R. P. SMEDLEYSTEVENSON, A. G. BUCHAN, and M. D. EATON, "Optimal Discontinuous Finite Element Methods for the Boltzmann Transport Equation with Arbitrary Discretisation in Angle", Ann. Nucl. Energy, 35, 1741 (2008).
27. J. H. JEANS, "The Equations of Radiative Transfer of Energy", Monthly Notices, Royal Astronomical Society, 78, 28 (1917).
28. S. CHANDRASEKHAR, Radiative Transfer, Clarendon Press, Oxford (1950).
29. D. KERSHAW and J. HARTE, "2D Deterministic Radiation Transport with the Discontinuous Finite Element Method", UCRL-ID-115525, Lawrence Livermore National Laboratory (1993).