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In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J.Comput. Phys., Vol. 255, 2013, pp 680-698] originally constructed for solvingthe BGK equation, to the more challenging case of the Boltzmann equation. Thescheme combines a robust and fast method for treating the transport part basedon an innovative Lagrangian technique supplemented with fast spectral schemesto treat the collisional operator by means of an operator splitting approach.This approach along with several implementation features related to theparallelization of the algorithm permits to construct an efficient simulationtool which is numerically tested against exact and reference solutions onclassical problems arising in rarefied gas dynamic. We present results up tothe $3$D$\times 3$D case for unsteady flows for the Variable Hard Sphere modelwhich may serve as benchmark for future comparisons between different numericalmethods for solving the multidimensional Boltzmann equation. For this reason,we also provide for each problem studied details on the computational cost andmemory consumption as well as comparisons with the BGK model or the limit modelof compressible Euler equations.

An efficient numerical method for solving the Boltzmann equation in multidimensions