This thesis deals with moment approximations for time-dependent linear kinetic transport equations, which arise for example in electron radiation therapy or radiative heat transfer problems. Moments are defined by angular averages against basis functions to produce spectral approximations in the angular variable. A typical family of moment models are the so-called PN methods, which are pure spectral methods. Another class of moment models is the family of minimum-entropy models, colloquially known as MN models or entropy-based moment closures, which has non-negative distribution functions (for certain physically relevant entropies). This is a desirable property since otherwise non-physical effects like non-negative densities may occur. However, the MN models suffer from the severe zero-netflux problem. For this reason, a new class of minimum-entropy models with mixed moments will be derived in general for one and two spatial dimensions.
All these properties of the minimum-entropy ansatz (for all types of angular bases) come at the price that the reconstruction of the distribution function involves solving a non-linear optimization problem at every point of the space-time mesh. One way to overcome this problem is to derive so-called Kershaw closures, which make use of the underlying realizability theory to obtain a realizable but analytically closed system. It turns out that in most cases Kershaw closures are also a good approximation to the corresponding minimum-entropy models. However, since it requires some knowledge about the realizable set, it is in general not possible to derive those closures for every type of angular basis.
Thus, in particular in higher dimensions, minimum-entropy models have to be considered. Despite the inherent parallelizability of the numerical optimization algorithm, the gain in efficiency that would come from a higher-order space-time discretization is necessary for a practical implementation. The key challenge for high-order methods for entropy-based moment closures is that the numerical solutions may leave the set of realizable moments, outside of which the defining optimization problem has no solution. Two classes of realizability-preserving high-order schemes (Runge-Kutta discontinuous-Galerkin and kinetic schemes) are derived and applied to the different moment models, effectively showing the efficiency of these high-order schemes.