
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 threedimensional 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 threedimensional $(r,\vartheta,z)$ 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 twopoint 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 twopoint systems of equations are solved by the cyclic throughcomputations 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 threedimensional 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 nononedimensional 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 threedimensional geometry, weighted diamond differencing scheme, $\mathrm{KP}_1$ acceleration scheme for inner iterations.
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Computational Mathematics and Mathematical Physics, 2009, 49:2, 334–362
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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 threedimensional 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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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.

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

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