Show Summary Details
More options …

# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.76

Online
ISSN
1609-9389
See all formats and pricing
More options …
Volume 16, Issue 4

# A Weakly Penalized Discontinuous Galerkin Method for Radiation in Dense, Scattering Media

Guido Kanschat
/ José Pablo Lucero Lorca
Published Online: 2016-07-06 | DOI: https://doi.org/10.1515/cmam-2016-0023

## Abstract

We review the derivation of weakly penalized discontinuous Galerkin methods for scattering dominated radiation transport and extend the asymptotic analysis to non-isotropic scattering. We focus on the influence of the penalty parameter on the edges and derive a new penalty for interior edges and boundary fluxes. We study how the choice of the penalty parameters influences discretization accuracy and solver speed.

MSC 2010: 65N30; 35Q20; 65N55; 82D75

## References

• [1]

Adams M. L., Discontinuous finite element transport solutions in thick diffusive problems, Nuclear Sci. Eng. 137 (2001), no. 3, 298–333. Google Scholar

• [2]

Asadzadeh M., Analysis of a fully discrete scheme for neutron transport in two-dimensional geometry, SIAM J. Numer. Anal. 23 (1986), 543–561. Google Scholar

• [3]

Asadzadeh M., Kumlin P. and Larsson S., The discrete ordinates method for the neutron transport equation in an infinite cylindrical domain, Math. Models Methods Appl. Sci. 2 (1992), no. 3, 317–338. Google Scholar

• [4]

Ayuso B. and Marini L. D., Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1391–1420. Google Scholar

• [5]

Bangerth W., Heister T., Heltai L., Kanschat G., Kronbichler M., Maier M. and Turcksin B., The deal.II library, version 8.3., Arch. Numer. Softw. 4 (2016), no. 100, 1–11. Google Scholar

• [6]

Bangerth W., Heister T. and Kanschat G., deal.II Differential Equations Analysis Library, Technical Reference, 8.0 edition, 2013. Google Scholar

• [7]

Bramble J. H., Multigrid Methods, Pitman Res. Notes Math. Ser. 294, Longman Scientific, Harlow, 1993. Google Scholar

• [8]

• [9]

Castillo P., Cockburn B., Perugia I. and Schötzau D., An a priori error estimate of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), no. 5, 1676–1706. Google Scholar

• [10]

• [11]

Dautray R. and Lions J.-L., Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6: Evolution Problems II, Springer, Berlin, 2000. Google Scholar

• [12]

Grella K., Sparse tensor phase space Galerkin approximation for radiative transport, SpringerPlus 3 (2014), no. 1, 3–230. Google Scholar

• [13]

Grella K. and Schwab C., Sparse tensor spherical harmonics approximation in radiative transfer, J. Comput. Phys. 230 (2011), no. 23, 8452–8473. Google Scholar

• [14]

Guermond J.-L. and Kanschat G., Asymptotic analysis of upwind DG approximation of the radiative transport equation in the diffusive limit, SIAM J. Numer. Anal. 48 (2010), no. 1, 53–78. Google Scholar

• [15]

Hackbusch W., Multi-Grid Methods and Applications, Springer, Berlin, 1985. Google Scholar

• [16]

Hackbusch W. and Probst T., Downwind Gauss-Seidel smoothing for convection dominated problems, Numer. Linear Algebra Appl. 4 (1997), no. 2, 85–102. Google Scholar

• [17]

Johnson C. and Pitkäranta J., Convergence of a fully discrete scheme for two-dimensional neutron transport, SIAM J. Numer. Anal. 20 (1983), 951–966. Google Scholar

• [18]

Johnson C. and Pitkäranta J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), 1–26. Google Scholar

• [19]

Kanschat G., Preconditioning methods for local discontinuous Galerkin discretizations, SIAM J. Sci. Comput. 25 (2003), no. 3, 815–831. Google Scholar

• [20]

Kanschat G., Block preconditioners for LDG discretizations of linear incompressible flow problems, J. Sci. Comput. 22 (2005), no. 1, 381–394. Google Scholar

• [21]

Kanschat G., Discontinuous Galerkin Methods for Viscous Flow, Deutscher Universitätsverlag, Wiesbaden, 2007. Google Scholar

• [22]

Kanschat G. and Ragusa J., A robust multigrid preconditioner for ${S}_{N}$DG approximation of monochromatic, isotropic radiation transport problems, SIAM J. Sci. Comput. 36 (2014), no. 5, 2326–2345. Google Scholar

• [23]

Larsen E. W., The asymptotic diffusion limit of discretized transport problems, Nuclear Sci. Eng. 112 (1992), 336–346. Google Scholar

• [24]

Larsen E. W. and Morel J. E., Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. II, J. Comput. Phys. 83 (1989), no. 1, 212–236. Google Scholar

• [25]

Larsen E. W., Morel J. E. and Miller, Jr. W. F., Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes, J. Comput. Phys. 69 (1987), no. 2, 283–324. Google Scholar

• [26]

LeSaint P. and Raviart P.-A., On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York (1974), 89–123. Google Scholar

• [27]

Mihalas D. and Weibel-Mihalas B., Foundations of Radiation Hydrodynamics, Dover, New York, 1984. Google Scholar

• [28]

Oxenius J., Kinetic Theory of Particles and Photons, Springer, Berlin, 1986. Google Scholar

• [29]

Pietro D. A. D., Ern A. and Guermond J.-L., Discontinuous Galerkin methods for anisotropic semidefinite diffusion with advection, SIAM J. Numer. Anal. 46 (2008), no. 2, 805–831. Google Scholar

• [30]

Pitkäranta J., A non-self-adjoint variational procedure for the finite-element approximation of the transport equation, Transp. Theory Stat. Phys. 4 (1975), no. 1, 1–24. Google Scholar

• [31]

Pitkäranta J., On the variational approximation of the transport operator, J. Math. Anal. Appl. 54 (1976), no. 2, 419–440. Google Scholar

• [32]

Pitkäranta J., Approximate solution of the transport equation by methods of Galerkin type, J. Math. Anal. Appl. 60 (1977), no. 1, 186–210. Google Scholar

• [33]

Pitkäranta J. and Scott R. L., Error estimates for the combined spatial and angular approximations of the transport equation for slab geometry, SIAM J. Numer. Anal. 20 (1983), no. 5, 922–950. Google Scholar

• [34]

Ragusa J., Guermond J.-L. and Kanschat G., A robust ${S}_{n}$-DG-approximation for radiation transport in optically thick and diffusive regimes, J. Comput. Phys. 231 (2012), no. 4, 1947–1962. Google Scholar

• [35]

Reed W. and Hill T., Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, 1973. Google Scholar

• [36]

Sanchez R. and Ragusa J., On the construction of Galerkin angular quadratures, Nuclear Sci. Eng. 169 (2011), 133–154. Google Scholar

• [37]

Toselli A. and Widlund O., Domain decomposition methods. Algorithms and theory, Springer Ser. Comput. Math. 34, Springer, Berlin, 2005. Google Scholar

• [38]

Widmer G., Hiptmair R. and Schwab C., Sparse adaptive finite elements for radiative transfer, J. Comput. Phys. 227 (2008), no. 12, 6071–6105. Google Scholar

• [39]

Xu J., Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. Google Scholar

Revised: 2016-05-13

Accepted: 2016-05-20

Published Online: 2016-07-06

Published in Print: 2016-10-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: GSC 220

The first author was supported by the Sino-German Science Center (grant id 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015. The second author was supported by the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (HGS MathComp), DFG grant GSC 220 in the German Universities Excellence Initiative.

Citation Information: Computational Methods in Applied Mathematics, Volume 16, Issue 4, Pages 563–577, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

Export Citation