Robustly Hyperbolic 1D Closures for Classical and Quantum Hydrodynamics

Abstract

The use of moment-closure methods for the prediction of non-equilibrium gas flows offers many modelling and numerical advantages over traditional methods. In particular, the maximum-entropy family of moment closures has been found to be very successful, offering models described by hyperbolic systems of balance laws. Unfortunately, most maximum- entropy closures cannot be expressed in closed form. The only affordable option is to propose approximations which are presently only available for a limited number of moments. Other Grad-based moment closures have been proposed. However, almost all of them suffer from questionable modelling assumption or numerical difficulties. Fortunately quadrature-based moment methods have been shown to produce accurate, globally hyperbolic models. The resulting moment systems maintain balance-law form and can be constructed to be Galilean invariant for arbitrarily large number of moments, making them an excellent alternative. The present work starts by reviewing key concepts of kinetic theory and moment closure, followed by a non-exhaustive litterature review of the main ideas developped in the last decades. Then three journal articles, constituting this thesis by contribution, are included. The first article shows the potential of moment closures to produce reliable quantum hydro- dynamic models. As an initial investigation, a maximum-entropy approximation was used to predict the time evolution of a Gaussian wave packet in phase space, allowing for a better understanding of the advantages and drawbacks of the method for situation where quantum effects are significant. The second paper investigates a new technique for the construction of globally hyperbolic closures. A new orthogonal-polynomial-based moment closure hierarchy (OPMC) covering the even-order closures is also presented. The accuracy of the new hierarchy is accessed using numerical solutions of Riemann problems along with strong shock-wave calculations. Finally, the applicability of this novel hierarchy to plasma physics is investigated in the third paper. Discretized solutions of the Vlasov-Poisson-BGK system of equations are compared against solutions of the moment systems for two canonical plasma problems featuring both the collisional and collisionless regimes.