A spectral-Lagrangian deterministic solver for the Boltzmann equation for rarefied gas flows is proposed. Numerical solutions are obtained for the flow across normal shock waves of pure gases and mixtures by means of a time-marching method. Operator splitting is used. The solution update is obtained as a combination of the operators for the advection (or transport) and homogeneous (or collision) problems. For the advection problem, the Finite volume method is considered. For the homogeneous

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... lem, a spectral-Lagrangian numerical method is used. The latter is based on the weak form of the collision operator and can be used with any type of cross-section model. The conservation of mass, momentum and energy during collisions is enforced through the solution of a constrained optimization problem. Numerical results are compared with those obtained by means of the DSMC method. Very good agreement is found for the whole range of free-stream Mach numbers being considered. For the pure gas case, a comparison with experimentally acquired density profiles is also performed, allowing for a validation of the spectral-Lagrangian solver. Possible applications of rarefied gas dynamics include the computation of the flowfield around spacecraft entering planetary atmospheres and the flow in hypersonic wind tunnels. Understanding rarefied gas effects in aerospace applications is important for an accurate calculation of the aerodynamic coefficients during the early phase of the entry of a space capsule into a planetary atmosphere, prediction of the heat flux experienced by ballutes during entry and descent, and also a correct interpretation of experimental measurements. Attempts to compute rarefied flows by means of a hydrodynamic description based on the Navier-Stokes equations give inaccurate results due to the failure of Newton's law for the stress tensor and Fourier's law for the heat flux vector in the rarefied regime [1, 2] . The Boltzmann equation provides a statistical description of dilute gaseous systems valid from the rarefied to the hydrodynamic regime [1, 2] . It describes the evolution of the species distribution function in the phase-space. Once the distribution function of each species known, it is possible to compute macroscopic observables such as density, hydrodynamic velocity and temperature by means of suitable moments. The computation of numerical solutions of the Boltzmann equation is not trivial. This is due to the integrodifferential nature of the equation. A further source of difficulty is the high dimensionality of the problem (numerical solutions must be sought in the phase-space). Stochastic-like solutions of the Boltzmann equation can be obtained by means of the Direct-Simulation-Monte-Carlo (DSMC) method [3, 4] . The former is a particle-based technique and has proven to be accurate [4] . However, it shares the drawbacks of stochastic methods, the main one being the presence of noise in the numerical results [3] . The former problem affects, in particular, the accuracy of the solution for low speed and unsteady flows. Parallel to the development of the DSMC method, deterministic numerical methods for the Boltzmann equation have been proposed. These comprise, among all, discrete velocity models [5, 6] and spectral methods [7, 8] . The main advantage of a deterministic method over the DSMC technique is that the numerical solution obtained is not affected by numerical noise. Deterministic methods can also be applied to flow problems in the hydrodynamic and transition regime, for which the use of the DSMC method becomes prohibitively expensive [9] . In the present work, an already existing spectral-Lagrangian solver for the Boltzmann equation for hardsphere gases [7, 10] is extended in order to deal with gas mixtures and account for more realistic collision cross-section models. Numerical solutions of the Boltzmann equation are obtained for the flow across normal shock waves of pure gases and mixtures. The key-part of the spectral-Lagrangian Boltzmann solver is the computational algorithm for the evaluation of the collision operator. The former is based on the weak form of the latter and can be used with any type of cross-section model. The conservation of mass, momentum and energy during collisions is enforced through the solution of a constrained optimization problem. Despite the formulation for internal energy excitation and relaxation is included in the numerical method, computational results are only shown for pure and mixture of monatomic gases. The purpose of the paper consists in the verification and validation of the spectral-Lagrangian Boltzmann solver through comparison with DSMC and experimental results, respectively. The paper is structured as follows. Section II introduces the physical model. The numerical method is described in detail in Sect. III. Computational results are presented and discussed in Sect. IV. Conclusions are outlined Sect. V. II. Physical modeling II.A. Simplifying assumptions and conventions The physical model used in the present work is based on the following assumptions and conventions [11]: 1. The gas mixture is composed of identical particles. 2. Particles have discrete internal energy levels: • The indices of energy levels are stored in the set I S = {1, . . . , N s }, N s being the number of species. • The mass of the species (energy level) i ∈ I S is m i . • The degeneracy and the energy of the energy level i ∈ I S are g i and E i , respectively.