First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

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.Schließlich beschäftigen wir uns mit der Möglichkeit einer zusätzlichen Modellreduktion für parameterabhängige Momentenmodelle. Hierzu nutzen wir die Reduzierte-Basis-Methode und berechnen mit der Hauptkomponentenanalyse ("Proper Orthogonal Decomposition", POD) ein reduziertes Modell. Wir stellen die hierarchische approximative POD (HAPOD) vor, ein universelles und einfach zu implementierendes Verfahren um eine approximative POD zu berechnen. Die HAPOD nutzt eine fast beliebig an die verfügbaren Ressourcen anpassbare Baumstruktur, so dass jeder Rechenknoten nur die POD einer relativ kleinen Teilmenge der Eingangsvektoren berechnen muss. Wir führen eine ausführliche theoretische Analyse der HAPOD durch und zeigen die Anwendbarkeit auf lineare Momentenmodelle sowie eine gegenüber der POD deutlich verbesserte Effizienz.

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

“…For non-smooth problems, however, instead of increasing the polynomial order N , it might be better to keep N fixed and regard piecewise polynomials on increasingly refined partitions of the domain. We will here restrict ourselves to piecewise linear bases (N = 1) which avoid many of the performance and realizability problems of the classical polynomial models [62,63]. In the following, we will shortly state the definitions of the first-order bases.…”

“…In the following, we will shortly state the definitions of the first-order bases. For a more detailed introduction see [63].…”

“…For Maxwell-Boltzmann entropy, it can be shown [33,63] that this set is equal to the positively realizable set .26) and that the map α b,η : R + b → R n given by (2.18) is a diffeomorphism with inverse map…”

“…Checking realizability is much easier when using piecewise linear bases instead of the standard polynomial basis on the whole velocity space [19,20,49,59,[62][63][64]. In addition, the computational cost is significantly lower for these models.…”

“…In addition, the computational cost is significantly lower for these models. However, solving the optimization problems is still costly compared to linear models and maintaining realizability still requires additional limiters [62].…”