Modeling and analysis of nonlinear closures in hierarchical moment equations using discontinuous Galerkin methods

The classical continuum mechanical equations of Navier-Stokes-Fourier (NSF) accurately describe gases with a high collision frequency, characterized by a small Knudsen number (Kn). However, the NSF equations fail for gases with a high Knudsen number, which corresponds to a low collision frequency. Solving Boltzmann's equation directly is a highly accurate approach, but it is computationally very demanding due to its high dimensionality. Another method is to approximate the Boltzmann equation by directly solving for macroscopic properties, such as density, velocity and temperature, which can be described by moments. Therefore, the so-called moment system is solved, consisting of an infinitely large system of hierarchically coupled partial differential equations (PDEs), with the moments as unknowns. Solving this system is impossible, as each moment depends on the next higher moment, which in turn depends on an even higher moment. Therefore, the moment system is truncated after M equations, requiring a moment closure for the (M+1) -th moment. This thesis discusses the extended Gramian closure, a recently developed moment closure based on orthogonal polynomials. The extended Gramian closure has attractive structure-preserving properties. It overcomes shortcomings of classical closures, such as the lack of global hyperbolicity in Grad's closure and the computational effort required by the maximum entropy closure. This work introduces the theoretical framework of the extended Gramian closure and suggests an updated closure under the condition that the equations are truncated where M is odd. Rigorous mathematical proofs for the structure-preserving properties of this updated closure are discussed. While the closure is constructed in one dimension, a projection method for applying it to a three-dimensional velocity space, based on matrix rotations, is introduced. Further, the extended Gramian closure is numerically approximated with the discontinuous Galerkin method, and its accuracy is analyzed and compared with that of the Gramian closure and Grad's closure. To this end, various benchmark problems, including the shock tube, shock structure, Landau damping and two-stream instability, are analyzed.