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Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 2, Pages 344–372 (Mi zvmmf45)  

This article is cited in 3 scientific papers (total in 3 papers)

$\mathrm{KP}_1$ acceleration scheme for inner iterations consistent with the weighted diamond differencing scheme for the transport equation in three-dimensional geometry

A. M. Voloshchenko

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047, Russia

Abstract: For the transport equation in three-dimensional $(r,\vartheta,z)$ geometry, a $\mathrm{KP}_1$ acceleration scheme for inner iterations that is consistent with the weighted diamond differencing (WDD) scheme is constructed. The $P_1$ system for accelerating corrections is solved by an algorithm based on the cyclic splitting method (SM) combined with Gaussian elimination as applied to auxiliary systems of two-point equations. No constraints are imposed on the choice of the weights in the WDD scheme, and the algorithm can be used, for example, in combination with an adaptive WDD scheme. For problems with periodic boundary conditions, the two-point systems of equations are solved by the cyclic through-computations method elimination. The influence exerted by the cycle step choice and the convergence criterion for SM iterations on the efficiency of the algorithm is analyzed. The algorithm is modified to threedimensional $(x,y,z)$ geometry. Numerical examples are presented featuring the $\mathrm{KP}_1$ scheme as applied to typical radiation transport problems in three-dimensional geometry, including those with an important role of scattering anisotropy. A reduction in the efficiency of the consistent $\mathrm{KP}_1$ scheme in highly heterogeneous problems with dominant scattering in non-one-dimensional geometry is discussed. An approach is proposed for coping with this difficulty. It is based on improving the monotonicity of the difference scheme used to approximate the transport equation.

Key words: transport equations in three-dimensional geometry, weighted diamond differencing scheme, $\mathrm{KP}_1$ acceleration scheme for inner iterations.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:2, 334–362

Bibliographic databases:

Document Type: Article
UDC: 519.634
Received: 27.02.2007
Revised: 30.05.2008

Citation: A. M. Voloshchenko, “$\mathrm{KP}_1$ acceleration scheme for inner iterations consistent with the weighted diamond differencing scheme for the transport equation in three-dimensional geometry”, Zh. Vychisl. Mat. Mat. Fiz., 49:2 (2009), 344–372; Comput. Math. Math. Phys., 49:2 (2009), 334–362

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. I. Kokonkov, O. V. Nikolaeva, “Consistent $\mathrm{P_1}$ Synthetic Acceleration of Inner Transport Iterations in $\mathrm{3D}$ Geometry”, Preprinty IPM im. M. V. Keldysha, 2015, 007, 28 pp.  mathnet
    2. N. I. Kokonkov, O. V. Nikolaeva, “An iterative $\mathrm{KP}_1$ method for solving the transport equation in $\mathrm{3D}$ domains on unstructured grids”, Comput. Math. Math. Phys., 55:10 (2015), 1698–1712  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. Panferov P., Kochkin V., Erak D., Makhotin D., Reshetnikov A., Timofeev A., “Changes To Irradiation Conditions of Vver-1000 Surveillance Specimens Resulting From Fuel Assemblies With Greater Fuel Height”, Isrd 15 - International Symposium on Reactor Dosimetry, Epj Web of Conferences, 106, ed. Lyoussi A., E D P Sciences, 2016, 02013  crossref  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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