Numerical Integrators for Physical Applications

Abstract

In this thesis, we report on our work in two very fundamental fields of physics which still have not been merged in a satisfactory way by a combining physical theory. One area is the field of very small particles, most accurately described by quantum mechanics. Here, we are interested in the phenomenon of superconductivity. The other area is that of the very heavy objects of our universe. Their most fundamental description is based on the theory of general relativity. Our particular interest lies in binary systems of compact objects rotating around each other, constantly radiating gravitational waves in the process. Although quantum mechanics and general relativity are worlds apart from a physical point of view, they inhibit some analogies when seen from our numerical perspective. For our aim is the same in both fields: We want to provide numerical tools for the simulations of interesting physical processes.

Regarding binary systems we want to compare two descriptions of their motion in space. The first is given by the Mathisson--Papapetrou equations. In order to study the evolution as given by these equations, we develop an efficient integration scheme based on Gauss Runge--Kutta methods. An intriguing challenge is given by the fact that part of the equations of motion have only be given implicitly. All obstacles notwithstanding, we present an efficient integrator which preserves the constants of motion even over long times. The second description of a binary's motion is given by a Hamiltonian approximation of the Mathisson--Papapetrou equations. We want to study whether this prescription yields physically valid results. To this aim, we first come up with an efficient numerical evolution scheme, again recurring to Gauss Runge-Kutta integrators. Our scheme conserves the Hamiltonian structure, thus yielding reliable results for long time spans. Then, we test the Hamiltonian approach in different aspects. When studying the behavior of important constants of motion, we have found out that the Hamiltonian in its originally published form must be based on unphysical assumptions. This triggered new theoretical studies by our collaborators from physics with the aim of finding better suited alternatives. Their new results and suggestions are tested with the help of our algorithms. The --now physically reasonable-- Hamiltonian descriptions are well-suited to investigate the binary systems for chaos with the help of surface sections. Hence, we take use of the collocation property of the Gauss Runge--Kutta schemes to present an accurate and convenient algorithm for the calculation of such sections.

In the realm of superconductivity, we consider the time-dependent BCS equations. These are quite involved partial differential equations describing the evolution of the Cooper pair density within a superconducting material or a superfluid. A very hot topic in the theoretical physics community concerns the question as to whether there exists, close to the critical temperature, a more convenient equation for a reliable approximation on a macroscopic scale. We take on this question from a numerical point of view. For this, we compare the evolution of a system with contact interaction given by the BCS equations to the one obtained via a linearized approximation by means of a thorough numerical study. We concentrate on a translation invariant system and develop two new numerical solvers based on so-called splitting methods. Splitting the coupled equations into more convenient subproblems and aptly combining the partial results, we come up with efficient and accurate schemes whose CPU times depend only linearly on the number of basis functions of the space discretization. With the help of the Fast Fourier Transform (FFT) algorithm, we can even extend our integrators to general potentials in a very natural way. In this case, too, the CPU effort grows only mildly as a function of the number of basis functions.

In the physically relevant case of a fermionic system interacting via a contact interaction, we employ our newly developed schemes to conduct numerous simulations for temperatures closer and closer to the critical one. From these simulations, we conclude that the linearization deviates far from the original equations. More precisely, the linear approximation leads to an exponential decay of the Cooper pair density whereas the full equations yield oscillations about a finite value. Consequently, the diffusion which is inherent to all hitherto existing macroscopic theories can only be an unphysical artifact. With this, we add an important fact to the still ongoing discussion in the physics community.

In short, we successfully developed convenient tools for the simulation of important physical phenomena in two fundamental fields of physics. This allowed our collaborators to gain valuable insights into the behavior of their equations of interest, thus contributing to the advance of fundamental science.