Model reduction for kinetic equations: moment approximations and hierarchical approximate proper orthogonal decomposition

Leibner Tobias

Abstract

Moment models are a class of specialized approximate models for kinetic transport equations. These models transform the kinetic equation to a system of equations for weighted velocity averages of the solution, called moments, thereby removing the velocity dependency. The properties of the resulting models depend on the chosen weight functions for the moments and on the approach used to close the equations. Closing the system by specifying a linear ansatz function results in linear models that are comparatively easy to solve but may show non-physical behaviour. Minimum-entropy moment closures, on the other hand, conserve many of the fundamental physical properties but require the solution of a non-linear optimization problem for every cell in the space-time grid. In addition, these models are only well-defined if the moments can be kept within a particular subset of the real coordinate space, the so-called realizable set, which is particularly challenging for higher-order numerical solvers. In this thesis, we investigate several approaches to increase usability of (minimum-entropy) moment approximations to linear kinetic equations. First, we focus on the basis functions, i.e. the weight functions for the velocity averages. Typically, these are chosen as polynomials on the whole velocity space. Here, as an alternative, we analyse models based on continuous and discontinuous piecewise linear basis functions. We show that, for non-smooth solutions, approximation quality is comparable while realizability conditions are much simpler and the computations are considerably faster. In addition, we provide second-order finite volume schemes and show that realizability preservation and thus overall implementation is significantly less complex for the new models. Second, we suggest a new numerical method for the minimum-entropy models that is based on a transformation of the moment vectors. This new method replaces the non-linear optimization problems by inversions of positive definite matrices. As a result, it avoids the realizability-related problems and is often several times faster than the standard optimization-based scheme. Finally, we investigate the possibility to further accelerate computations for parameter-dependent moment models by building a reduced model via the reduced basis method utilizing proper orthogonal decomposition (POD). We present the hierarchical approximate POD (HAPOD), a general, easy-to-implement approach to compute an approximate POD based on arbitrary tree hierarchies of worker nodes, where each worker computes a POD of only a small number of input vectors. The tree hierarchy can be freely adapted to optimally suit the available computational resources. We provide rigorous theoretical results and show applicability and performance by applying the HAPOD to a large linear moment model.