A novel multiproblem methodology devised to manufacture highly accurate numerical solutions of the linear Boltzmann equation is proposed. As an alternative to classical discretization schemes that focus on a single mesh, the multiproblem approach seeks transport solutions as the limit of a sequence of calculations executed on successively more refined grids. The sequence of approximations serves as a basis for the extrapolation of the solution toward its mesh-independent limit. Furthermore, the multiproblem strategy allows an optimization of the computational effort whenever compared to the single-grid approach. Indeed, the solution obtained on an unrefined mesh is employed as the starting guess for transport calculations on the next grid of the sequence, drastically reducing the number of inner iterations needed on the highly refined mesh. The efficiency of the algorithm may be further improved by combining the source iterations with a convergence acceleration scheme based on nonlinear extrapolation algorithms. To evaluate the performance of the proposed approach, the multiproblem methodology is applied to solve linear transport problems in spherical geometry, which are known to feature special properties whenever compared with the transport of particles in Cartesian geometry. The methodology is implemented by choosing the presumably simplest and most widespread numerical transport algorithm (i.e., discrete ordinates with diamond differences). Results show that five- to six-digit accuracy can be obtained in a competitive computational time without resorting to powerful workstations.