A $KP_1$ scheme for acceleration of inner iterations for the transport equation in 3D geometry consistent with nodal schemes: 2. Splitting method for solving the $P_1$ system for acceleration corrections

An algorithm is proposed for solving the ${{P}_{1}}$ system for acceleration corrections that arises in constructing a $K{{P}_{1}}$ scheme for accelerating the convergence of inner iterations consistent with the nodal LD (Linear Discontinues) and LB (Linear Best) schemes of third and fourth-order accuracy in space for the transport equation in three-dimensional $r,\vartheta,z$ geometry. The algorithm is based on a cyclic splitting method combined with the through-computation algorithm for solving auxiliary two-point equations system. A modification of the algorithm is considered for three-dimensional $x,y,z$ geometry.