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

A. M. Voloshchenko

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 2, pp. 344-372

Résumé

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.