A $K{{P}_{1}}$ scheme for acceleration of inner iterations for the transport equation in 3D geometry consistent with nodal schemes: basic equations and numerical results

A $K{{P}_{1}}$ scheme for accelerating the convergence of inner iterations for the transport equation in three-dimensional $r,\vartheta ,z$ geometry is constructed. 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 cyclic splitting method (SM) combined with the tridiagonal matrix algorithm to solve auxiliary systems of two-point equations. A modification of the algorithm for three-dimensional $x,y,z$ geometry is considered. Numerical examples of using the $K{{P}_{1}}$ scheme to solve typical radiation transport problems in three-dimensional geometries are given, including problems with a significant role of scattering anisotropy and highly heterogeneous problems with dominant scattering.