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Mathematics of Computation

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An adaptive nested source term iteration for radiative transfer equations

Authors: Wolfgang Dahmen, Felix Gruber and Olga Mula
Journal: Math. Comp. 89 (2020), 1605-1646
MSC (2010): Primary 65N12, 65N15, 65N30; Secondary 65N50
Published electronically: January 8, 2020
Uncorrected version: Original version posted January 8, 2020
Corrected version: This updated version of the article fixes a publisher error in cross-linking
MathSciNet review: 4081913
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Abstract: We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds for the $L_2$ norm. A key role is played by stable Petrov–Galerkin-type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite-dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of the iteration. To ensure that the stage dependent error tolerances are met, one has to employ rigorous a posteriori error bounds which, in our case, rest on a Discontinuous Petrov–Galerkin (DPG) scheme. These a posteriori bounds are not only crucial for guaranteeing the convergence of the perturbed iteration but are also used to generate adapted parameter dependent spatial meshes. This turns out to significantly reduce overall computational complexity. Since the global operator is only applied, we avoid the need to solve linear systems with densely populated matrices. Moreover, the approximate application of the global scatterer is accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with non-trivial scattering kernels.

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Additional Information

Wolfgang Dahmen
Affiliation: Mathematics Department, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29208
MR Author ID: 54100

Felix Gruber
Affiliation: IGPM, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany

Olga Mula
Affiliation: CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University, 75016 Paris, France
MR Author ID: 1027794

Keywords: DPG transport solver, iteration in function space, fast application of scattering operator, Hilbert–Schmidt decomposition, matrix compression, a posteriori bounds, kinetic problems, linear Boltzmann, radiative transfer
Received by editor(s): November 20, 2018
Received by editor(s) in revised form: October 3, 2019, and October 22, 2019
Published electronically: January 8, 2020
Additional Notes: The research of the first author was supported by the NSF Grant DMS 1720297, and by the SmartState and Williams-Hedberg Foundation.
The third author is the corresponding author
Article copyright: © Copyright 2020 American Mathematical Society