# A conservative deterministic spectral method for rarefied gas flows

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## Abstract

The mathematical analysis of the Boltzmann equation for a wide range of important models is well developed. It describes physical phenomena which are often of great engineering importance (in aerospace industry, semiconductor design, etc.). For that reason, analytical and computational methods of solving the Boltzmann equation are studied extensively. The idea of describing processes on a scale of the order of the relaxation scales of time and space has been realized. The nonlinear Boltzmann equation possesses the important essence of a physically realistic equation, so it is possible not only to consider the flows of simple media but to formulate new problems due to the ability of this equation to describe nonequilibrium states. In this dissertation, a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation for variable hard potential (VHP) collision kernels with conservative or non-conservative binary interactions is proposed. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in the collision integral computation is reduced to a separate integral over the unit sphere S2. In addition, the conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or to elastic model of slowing down processes. We prove the accuracy, consistency and conservation properties of the proposed conservative spectral method. Existing spectral methods have consistency proofs which are only for elastic collisions, and also such methods do not conserve all the necessary moments of the collision integral. In this dissertation, error estimates for the conservation routine are provided. Such conservation correction is implemented as an extended isoperimetric problem with the moment conservation properties as the constraints. We use and extend an existing bound estimate of Gamba, Panferov and Villani for the inelastic/elastic space homogeneous Boltzmann collision operator. The result is an original extension to the work of Gustaffson. Using these estimates along with projection error estimates and conservation correction estimates, we prove that the conservation correction is bounded by the spectral accuracy. Simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation for both elastic and inelastic VHP interactions. Benchmarking of the self-similar simulations involves the selection of a time rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard-spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods. Recognizing the importance of the Boltzmann equation in the analysis of shock structures and nonequilibrium states, such a study is done for 1D(x) × 3D(v). The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Sone and Aoki, where the wall temperature is suddenly changed, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock.