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A new discrete velocity scheme for solving the Boltzmann equation has been implemented for homogeneous relaxation and one-dimensional problems. Directly solving the Boltzmann equation is computationally expensive because in addition to working in physical space, the nonlinear collision integral must also be evaluated in a velocity space. To best solve the collision integral, collisions between each point in velocity space with all other points in velocity space must be considered, but this is very expensive. Motivated by the Direct Simulation Monte Carlo (DSMC) method, the computational costs in the present method are reduced by randomly sampling a set of collision partners for each point in velocity space. A collision partner selection algorithm was implemented to favor collision partners that contribute more to the collision integral. The new scheme has a built in flexibility, where the resolution in approximating the collision integral can be adjusted by changing how many collision partners are sampled. The computational cost associated with evaluation of the collision integral is compared to the corresponding statistical error. Having a fixed set of velocities can artificially limit the collision outcomes by restricting post collision velocities to those that satisfy the conservation equations and lie precisely on the grid. A new velocity interpolation algorithm enables us to map velocities that do not lie on the grid to nearby grid points while preserving mass, momentum, and energy. This allows for arbitrary post-collision velocities that lie between grid points or completely outside of the velocity space to be projected back onto the nearby grid points. The present scheme is applied to homogeneous relaxation of the non-equilibrium Bobylev Krook-Wu distribution, and the numerical results agree well with the analytic solution. After verifying the proposed method for spatially homogeneous relaxation problems, the scheme was then used to solve a 1D traveling shock. The jump conditions across the shock match the Rankine-Hugoniot jump conditions. The internal shock wave structure was then compared to DSMC solutions, and good agreement was found for Mach numbers ranging from 1.2 to 6. Since a coarse velocity discretization is required for efficient calculation, the effects of different velocity grid resolutions are examined. Although using a relatively coarse approximation for the collision integral is computationally efficient, statistical noise pollutes the solution. The effects of using coarse and fine approximations for the collision integral are examined and it is found that by coarsely evaluating the collision integral, the computational time can be reduced by nearly two orders of magnitude while retaining relatively smooth macroscopic properties. / text

<p>Comets are the most primordial and least evolved bodies in the Solar System. As such, they are unique sources of information regarding the early history of the Solar System. However, little is known about cometary nuclei since they are very difficult to observe due to the obscuring coma. Indirect methods are therefore often used to extract knowledge about nucleus parameters such as size, shape, density, material strength, and rotational properties. For example, tidal and non-tidal splitting of cometary nuclei can provide important information about nuclear densities and material strengths, but only if the criteria for mechanical stability are well known. Masses and densities of cometary nuclei can also be obtained by studying orbital modifications due to non-gravitational forces, but only if the thermophysics of comets can be modelled accurately. </p><p>A detailed investigation is made regarding the mechanical stability of small Solar System bodies. New expressions for the Roche distance are derived, as functions of the size, shape, density, material strength, rotational period, and spin axis orientation of a body. The critical rotational period for centrifugal breakup in free space is also considered, and the resulting formulae are applied to comets for which the size, shape and rotational period have been estimated observationally, in order to place constraints on their densities and material strengths. </p><p>A new thermophysical model of cometary nuclei is developed, focusing on two rarely studied features - layer absorption of solar energy, and parallel modelling of the nucleus and innermost coma. Sophisticated modelling of radiative transfer processes and the kinetics of gas in thermodynamic non-equilibrium form the basis for this work. The new model is applied to Comet 19P/Borrelly, and its density is estimated by reproducing the non-gravitational changes of its orbit.</p>

Comets are the most primordial and least evolved bodies in the Solar System. As such, they are unique sources of information regarding the early history of the Solar System. However, little is known about cometary nuclei since they are very difficult to observe due to the obscuring coma. Indirect methods are therefore often used to extract knowledge about nucleus parameters such as size, shape, density, material strength, and rotational properties. For example, tidal and non-tidal splitting of cometary nuclei can provide important information about nuclear densities and material strengths, but only if the criteria for mechanical stability are well known. Masses and densities of cometary nuclei can also be obtained by studying orbital modifications due to non-gravitational forces, but only if the thermophysics of comets can be modelled accurately. A detailed investigation is made regarding the mechanical stability of small Solar System bodies. New expressions for the Roche distance are derived, as functions of the size, shape, density, material strength, rotational period, and spin axis orientation of a body. The critical rotational period for centrifugal breakup in free space is also considered, and the resulting formulae are applied to comets for which the size, shape and rotational period have been estimated observationally, in order to place constraints on their densities and material strengths. A new thermophysical model of cometary nuclei is developed, focusing on two rarely studied features - layer absorption of solar energy, and parallel modelling of the nucleus and innermost coma. Sophisticated modelling of radiative transfer processes and the kinetics of gas in thermodynamic non-equilibrium form the basis for this work. The new model is applied to Comet 19P/Borrelly, and its density is estimated by reproducing the non-gravitational changes of its orbit.

En physique du transport, en particulier en physique du transfert radiatif, la méthode de Monte-Carlo a été développée à l'origine comme la simulation de l'histoire d'un grand nombre de particules, dont on déduit des observables moyennes. Cette méthode numérique doit son succès à plusieurs qualités : une gestion naturelle des espaces des phases aux nombreuses dimensions, une erreur systématique nulle par rapport au modèle physico-mathématique, les intervalles de confiance donnés avec les résultats, une capacité à prendre en compte simultanément de nombreux phénomènes physiques, la possibilité de calcul de sensibilités simultané, et une parallélisation aisée. En cinétique des gaz, les particules collisionnent entre elles et non pas avec un milieu extérieur ; on dit que leur transport est non-linéaire. Ces collisions mutuelles mettent en défaut l'approche évoquée ci-dessus de la méthode de Monte-Carlo ; car pour simuler des trajectoires indépendantes de multiples particules et ainsi estimer leur distribution, il faut connaître au préalable exactement cette même distribution...Cette thèse fait suite à celles de Jérémi DAUCHET (2012) et de Mathieu GALTIER (2014), consacrées au transfert radiatif. Entre autres travaux, ces auteurs montraient comment la méthode de Monte-Carlo peut s'accommoder de non-linéarités, en gardant son formalisme et ses spécificités habituelles. Les non-linéarités alors franchies étaient respectivement une loi de couplage chimie/luminance, et la dépendance de la luminance envers le coefficient d'absorption. On essaie dans ce manuscrit d'outrepasser la non-linéarité du transport. Pour cela, nos principaux outils sont un suivi des particules en remontant le temps, basé sur des formulations intégrales des équations de transport, formulations largement inspirées des algorithmes dits à collisions nulles. Nous montrons, sur plusieurs exemples académiques, que nous avons en effet étendu la méthode de Monte-Carlo à la résolution de l'équation de Boltzmann. Ces exemples sont aussi l'occasion de tester les limites de ce que nous avons mis en place. Les résultats les plus marquants sont certainement l'absence totale de maillage dans la méthode numérique, ainsi que sa capacité à calculer correctement les quantités de particules de haute énergie cinétique (toujours peu nombreuses par rapport au total, en cinétique des gaz). Au-delà des exemples fournis, ce manuscrit est voulu comme un essai de formalisme et une exploration des bases de la méthode développée. L'accent est mis sur les raisonnements menant à la mise au point de la méthode, plutôt que sur les implémentations particulières qui ont été abouties. La méthode est encore, aux yeux de l'auteur, largement susceptible d'être retravaillée. En particulier, les temps maximaux sur lesquels l'évolution des particules est calculable, qui constituent la faiblesse principale de la méthode numérique développée, peuvent sûrement être augmentés. / In transport physics, especially in radiative transfer physics, the Monte-Carlo method has been originally developed as the simulation of the history of numerous particles, from which are deduced mean observables. This numerical method owes its success to several qualities : a natural management of many-dimensional phase space, a null systematic error away from the mathematical and physical model, the confidence intervals given with the results, an ability to take into account simultaneously numerous physical phenomenons, the simultaneous sensitivities calculating possibility, and an easy parallelization. In gas kinetics, particles collide each other, not with an external fixed medium ; it is said that their transport is non-linear. These mutual collisions put out of action the aforesaid approach of the Monte-Carlo method ; because in order to simulate the independent trajectories of multiple particles and thus estimate their distribution, this distribution must beforehand be exactly known...This thesis follows on from those of Jérémy DAUCHET (2012) and of Mathieu GALTIER (2014), dedicated to radiative transfer physics. Between other works, these authors have shown how the Monte-Carlo method can bear non-linearities, while keeping its customary formalism and specificities. The then overcome non-linearities were respectively a chemistry/irradiance coupling law, and the dependence of the irradiance toward the absorption coefficient. We try in this manuscript to overcome the non-linearity of the transport. In this aim, our main tools are a reverse following of particles, based on integral formulations of the transport equations, formulations largely inspired from the so-called null collisions algorithms. We show, on several academic examples, that we have indeed extended the Monte Carlo method to the resolution of the Boltzmann equation. These examples are also occasions to test the limits of what we have built. The most noteworthy results are certainly the absence of any mesh in the numerical method, and its capacity to calculate correctly the high-speed particles quantities (always rare compared to the total, in gas kinetics). Beyond the given examples, this manuscript is wanted as a formalism attempt and an exploration of the developed method basics. The focus is made on the reasoning leading to the method, rather than on particular implementations which have been realized. In the eyes of the author, the method is still largely reworkable. In particular, the maximal times on which the evolution of particles is computable, which constitute the main weakness of the developed numerical method, can surely be increased.