Entropy-based moment closures for rarefied gases and plasmas

Schärer, Roman Pascal; Torrilhon, Manuel (Thesis advisor); Macdonald, James (Thesis advisor)

Aachen (2016, 2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2016

Abstract

The method of moments provides a flexible mathematical framework to derive reduced-order models for the approximation of the kinetic Boltzmann equation. This thesis investigates moment equations based on the principle of entropy maximization to describe moderately rarefied gas and plasma flows in the transition regime between the collision dominated continuum and the free molecular flow regime.The maximum-entropy system has favorable mathematical features: The resulting system of partial differential equations is in conservative form, satisfies an H-theorem and is symmetric hyperbolic. However, the robust and efficient numerical solution of entropy-based moment closures is challenging: First, the maximum-entropy closure can have a singularity in the closing flux around the equilibrium distribution, rendering initial value problems with data in local thermodynamic equilibrium questionable. Second, the Hessian matrix of the Newton method used in the dual minimization problem for the Lagrange parameters is arbitrarily ill-conditioned. Third, moments of the maximum-entropy distribution function are in general not available in closed form and have to be evaluated numerically, which can result in excessive run times.The first issue can be avoided by bounding the velocity domain. Numerical examples show that enlarging the velocity domain leads to smaller sub-shocks, i.e., unphysical discontinuities in the continuous shock-structure problem. To study the effect of the singularity in the closing flux, a closed-form closure for a simplified toy model problem is considered. Numerical results demonstrate that the sub-shock in the continuous shock-structure problem can be mitigated by non-linear closures and eventually removed by a singularity in the closing flux.Several numerical examples are provided for the 35-moment system in slab geometry, showing promising results for shock-structure and Riemann problems. The use of an adaptive basis method for the dual minimization problem allows the robust simulation of strongly nonequilibrium processes.To reduce the excessive computational run times of the maximum-entropy closure, high-performance implementations of the numerical integration algorithms are developed for multi-core processors and graphics cards. Additionally, new efficient explicit and semi-implicit time-integration schemes based on a formulation in the Lagrange parameters of the dual minimization problem are presented.