The Boltzmann equation in 1D spherical geometry presents features that are generally encountered in the solution of particle transport in curvilinear geometry. The present paper describes a possible method to obtain highly accurate results by progressively building the solution from lower-order intermediate discretization schemes. Results prove the efficiency of this approach in terms of reduction of inner iterations and computational time. The concept of convergence acceleration can be suitably introduced into the algorithm to improve its performance and to speed up the inner iteration cycle. Finally, the application of extrapolation to the sequence of partial solutions is applied to seek the mesh-independent limiting solution. Several results, for both source and critical problems, are reported and compared to established benchmarks.