Efficient Monte Carlo description of multi-phase and multi-scale fluid flows in kinetic theory

In this doctoral thesis, efficient particle methods describing phase transition and multi-scale fluid flow far from equilibrium are devised. In the first part of this thesis, the short and long-range interactions are modeled through a continuous stochastic process and a Poisson-type partial differential equation, respectively, which allows the use of efficient numerics in practice. In particular, a Fokker-Planck type equation is devised to model the transition of probability measure associated with the underlying jump process of Enskog equation. Hence, as the main advantage over the direct Monte Carlo solution algorithms, the cost of velocity evolution can be decoupled from density and temperature since the numerical cost of It\={o} process associated with Fokker-Planck equation scales only with the number of computational particles. Furthermore, an efficient solution algorithm approximating the exact Vlasov integral, i.e., a kernel that describes long-range interactions of dense gases and liquids at moderate densities, was devised. The main idea is to relate the long-range molecular potential to the fundamental solution of a well-posed elliptic partial differential equation, which allows the use of efficient Poisson solvers. Therefore, the challenges associated with the numerics of local, long-tail, and high dimensional Vlasov integral for the interior of the simulation domain is avoided by introducing an accurate global solution at a lower cost. The devised models were tested against the benchmark in several test cases, such as the lid-driven cavity, Couette flow, evaporation to vacuum, and inverted temperature gradients. Reasonable accuracy at lower cost was attained in comparison with the direct Monte Carlo solution algorithm.In the second part of this thesis, the coupling of continuum conservation laws with a kinetic description of fluid flows far from the equilibrium is investigated. The mapping from one scale to another raises several challenges. Although the moments of distribution can be estimated using samples simply by the law of large numbers subject to a statistical noise, the inverse is ill-posed, which can be resolved by restricting the solution to the one that minimizes Shannon entropy, i.e., maximum entropy distribution (MED). The numerical challenges in the constrained optimization problem of MED motivated this research. As the first step, a high-dimensional regression method based on Gaussian process is devised to provide an accurate and efficient estimate of MED for a desired space of moments. Then, an all particle solution algorithm is devised to couple approximated solution of the Navier-Stokes-Fourier system of equations, e.g., Smoothed-Particle-Hydrodynamics, with the one for the Boltzmann equation, e.g., direct simulation Monte Carlo. The trained Gaussian process was tested in predicting the exact MED solution in several cases, such as the relaxation of distribution to equilibrium governed by Boltzmann equation in a homogeneous setting. Furthermore, the devised hybrid solution algorithm was tested by simulating the Sod's shock tube. Overall, reasonable accuracy compared with the benchmark with a speedup of two orders of magnitude was obtained