A $KP_1$ scheme for acceleration of inner iterations for the transport equation in two-dimensional geometries consistent with nodal schemes

A $KP_1$ scheme for accelerating the convergence of inner iterations is constructed for the transport equation in 2D $r$, $z$ geometry. This scheme is consistent with the nodal LD (Linear Discontinues) and LB (Linear Best) schemes of the third and fourth orders of accuracy with respect to the spatial variables. To solve the $P_1$ system for acceleration corrections, an algorithm is proposed based on the splitting method combined with the tridiagonal matrix algorithm to solve auxiliary systems of two-point equations. A modification of the algorithm for 2D $x$, $z$ geometry is considered. Numerical examples of using the $KP_1$ scheme to solve radiation transport problems in 2D $r$, $z$, $x$, $z$ and $r$, $\vartheta$ geometries are given, including problems with a significant role of scattering anisotropy and highly heterogeneous problems with dominant scattering.