Master Thesis

Magisterarbeit / master thesis

Strong Conservation Form and Grid Generation in Nonsteady Curvilinear Coordinates – Numerical Methods for Implicit Radiation Hydrodynamics in 2D and 3D

Motivation

In astrophysics a multitude of systems and configurations are described with concepts from hydrodynamics coupled with gravitation, radiation and magnetism. Mathematically radiation hydrodynamics (RHD) and magnetohydrodynamics (MHD) are systems of coupled nonlinear partial differential equations. Multiple fields in astrophysics have been adopting and developing sophisticated numerical methods for solving these PDEs in various applications. Explicit numerical schemes in computational fluid dynamics have been popular for many years and received substantial boosts due to advances in technology and parallelizing techniques. Riemann solvers and related methods are extensively adapted in 2D and 3D computations however they have one main disadvantage. An inherent limitation for time steps in explicit schemes impedes applications where various time scales are of interest. Hence a majority of these calculations emphasize on small temporal and spatial scales but fail to treat phenomenons properly on very diverse scales. Techniques for implicit numerics are computationally more expensive since they are hardly parallelizable but do not limit the increments of the scheme intrinsically. Implicit RHD in 1D has proven favorable for describing astrophysical processes on large spatial and temporal scales like nonlinear pulsation or long term development of supernovae. The idea of this thesis was to study suitable generalizations of these concepts to 2D and 3D. Additional degrees of freedom in multi-dimensional implicit RHD would disclose the treatment of large scale convection and how it interacts with rotation, mixing, the coupling of rotation and pulsation, transport of angular momentum and mass loss, interactions of discs and stars during star formation processes etc. The target of this thesis was to find the strong conservation form of the equations of radiation hydrodynamics in non-steady curvilinear coordinates and to study multi-dimensional adaptive grid generation.

Synopsis

The Euler equations of hydrodynamics, the Maxwell equations as well as radiative transport equations are hyperbolic PDEs that connect certain densities and fluxes via conservation laws. Generally they emerge from natural symmetries that constitute major principles in mathematical physics. The numerical implementation of these equations essentially needs to comprise these qualities. Applied mathematics have developed an articulate framework of numerical methods for conservation laws that ensure the conservation of mass, momentum, energy etc. if applied properly. The main  challenge is to compute the fluxes correctly which indeed will be the issue of the first three chapters in this paper. Self gravitation is described via the Poisson equation that poses an extra challenge due to its elliptic nature. While 1D computations avoid solving this pure boundary value problem by considering an integrated form we will have to design a different approach to self gravitation in 2D and 3D. A generalization of implicit conservative numerics to multiple dimensions requires advanced concepts of tensor analysis and differential geometry and hence a more thorough dedication to mathematical fundamentals than maybe expected at first glance. Hence we begin to discuss fundamental mathematics and physics of RHD with special focus on differential geometric consistency and study numerical methods for nonlinear conservation laws to gain a solid definition of the term conservative. The efforts in tensor analysis will be needed when applying Vinokurs theorem to gain the strong conservation form for conservation laws in general curvilinear coordinates. Moreover, it will be required to slightly reformulate the artificial viscosity for such nonlinear coordinates.
Astronomical objects are characterized by fast flows and high propagation speeds on the one hand but astronomical length and time scales on the other hand. Implicit numerical schemes are not affected by the Courant Friedrichs Levy condition which limits explicit schemes to rather impracticably small time steps. Implicit methods however produce algebraic problems that require matrix inversion which is computationally expensive. In order to achieve viable resolution, adaptive grid techniques have been developed. It is desired to treat processes on small length scales like shocks and ionization fronts as well as physics at the extent of the objects dimension itself like large scale convection flows and pulsations. The combination of implicit schemes and adaptive grids allows to
resolve astrophysics appropriately at various scales. In the last chapter of this paper we study problem oriented adaptive grid generation in 2D and 3D. We establish three main postulations for an ideal grid and analyze several feasible approaches.