Discontinuous Galerkin for the Radiative Transport Equation
This note presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed.
KeywordsFinite elements Discontinuous Galerkin Neutron transport Diffusion limit
This material is based upon work supported by the Department of Homeland Security under agreement 2008-DN-077-ARI018-02, National Science Foundation grants DMS-0713829, DMS-0810387, and CBET-0736202, and is partially supported by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
- M. L. Adams. Discontinuous finite element methods in thick diffusive problems. Nucl. Sci. Eng., 137:298–333, 2001.Google Scholar
- S. Chandrasekhar. Radiative Transfer. Oxford University Press, 1950.Google Scholar
- R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5. Evolution problems, I. Springer-Verlag, Berlin, Germany, 1992. ISBN 3-540-50205-X; 3-540-66101-8.Google Scholar
- A. Ern and J.-L. Guermond. Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM J. Numer. Anal., 44(2): 753–778, 2006.Google Scholar
- E. W. Larsen. Deterministic transport methods. Technical Report LA-9533-PR, Los Alamos Scientific Laboratory, Los Alamos, NM, 1982.Google Scholar
- E. W. Larsen. On numerical solutions of transport problems in the diffusion limit. Nucl. Sci. Engr., 83:90–99, 1983.Google Scholar
- E. W. Larsen and J. E. Morel. Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. II. J. Comput. Phys., 83(1):212–236, 1989.Google Scholar
- P. Lesaint and P.-A. Raviart. On a finite element method for solving the neutron transport equation. In Mathematical Aspects of Finite Elements in Partial Differential Equations, pages 89–123. Publication No. 33. Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974.Google Scholar
- W. Reed and T. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.Google Scholar