Dichtematrix-Renormierung, angewandt auf nichtlineare dynamische Systeme

Bogner T. Density matrix renormalisation applied to nonlinear dynamical systems. Bielefeld (Germany): Bielefeld University; 2007.In dieser Dissertation wird die effektive numerische Beschreibung nichtlinearer dynamischer Systeme untersucht. Systeme dieser Art tauchen praktisch überall auf, wo zeitabhängige Größen quantitativ untersucht werden, d.h. in fast allen Bereichen der Physik, aber auch in der Biologie, Ökonomie oder Mathematik. Ziel ist die Bestimmung reduzierter Modelle, deren Phasenraum eine signifikant reduzierte Dimensionalität aufweist. Dies wird erreicht durch Benutzung von Konzepten aus der Dichtematrix-Renormierung. In dieser Arbeit werden drei neue Anwendungen vorgeschlagen. Zuerst wird eine Dichtematrix-Renormierungsmethode zur Berechnung einer Schur-Zerlegung vorgestellt. Verglichen mit bereits existierenden Arbeiten liegt der Vorteil dieses Ansatzes in der Möglichkeit, auch für nicht-normale Operatoren orthonormale Basen von sukzessive invarianten Unterräumen zu bestimmen. Der Algorithmus wird dann angewandt auf Gittermodelle stochastischer Systeme, wobei als Beispiele ein Reaktions-Diffusions- und ein Oberflächenablagerungs-Modell dienen. Als Nächstes wird ein Dichtematrix-Renormierungsansatz für die orthogonale Zerlegung (proper orthogonal decomposition) entwickelt. Diese Zerlegung erlaubt die Bestimmung relevanter linearer Unterräume auch für nichtlineare Systeme. Durch die Verwendung der Dichtematrix-Renormierung werden alle Berechnungen nur für kleine Untersysteme durchgeführt. Dabei werden diskretisierte partielle Differentialgleichungen, d.h. die Diffusionsgleichung, die Burgers-Gleichung und eine nichtlineare Diffusionsgleichung als numerische Beispiele betrachtet. Schließlich wird das vorige Konzept auf höherdimensionale Probleme in Form eines Variationsverfahrens erweitert. Dies Verfahren wird dann an den zweidimensionalen Navier-Stokes-Gleichungen erprobt.In this work the effective numerical description of nonlinear dynamical systems is investigated. Such systems arise in most fields of physics, as well as in mathematics, biology, economy and essentially in all problems for which a quantitative description of a time evolution is considered. The aim is to find reduced models with a phase space of significantly reduced dimensionality. This is achieved by the use of concepts from density matrix renormalisation. Three new applications are proposed in this work. First, a density matrix renormalisation method for calculating a Schur decomposition is introduced. The advantage of this approach, compared to existing work, is the possibility to obtain orthonormal bases for successively invariant subspaces even if the generator of evolution is not normal. The algorithm is applied to lattice models for stochastic systems, namely a reaction diffusion and a surface deposition model. Next, a density matrix renormalisation approach to the proper orthogonal decomposition is developed. This allows the determination of relevant linear subspaces even for nonlinear systems. Due to the use of density matrix renormalisation concepts, all calculations are done on small subsystems. Here discretised partial differential equations, i.e. the diffusion equation, the Burgers equation and a nonlinear diffusion equation are considered as numerical examples. Finally, the previous concept is extended to higher dimensional problems in a variational form. This method is then applied to the two-dimensional, incompressible Navier-Stokes equations as testing ground

StaRMAP - A second order staggered grid method for spherical harmonics moment equations of radiative transfer

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent $P_N$ and $SP_N$ equations, of radiative transfer. The method, which works for arbitrary moment order $N$, makes use of the specific coupling between the moments in the $P_N$ equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.Comment: 28 pages, 7 figures; StaRMAP code available at http://www.math.temple.edu/~seibold/research/starma

Efficient computations for multiphase flow problems using coupled lattice Boltzmann-level set methods

Multiphase flow simulations benefit a variety of applications in science and engineering as for example in the dynamics of bubble swarms in heat exchangers and chemical reactors or in the prediction of the effects of droplet or bubble impacts in the design of turbomachinery systems. Despite all the progress in the modern computational fluid dynamics (CFD), such simulations still present formidable challenges both from numerical and computational cost point of view. Emerging as a powerful numerical technique in recent years, the lattice Boltzmann method (LBM) exhibits unique numerical and computational features in specific problems for its ability to detect small scale transport phenomena, including those of interparticle forces in multiphase and multicomponent flows, as well as its inherent advantage to deliver favourable computational efficiencies on parallel processors. In this thesis two classes of LB methods for multiphase flow simulations are developed which are coupled with the level set (LS) interface capturing technique. Both techniques are demonstrated to provide high resolution realizations of the interface at large density and viscosity differences within relatively low computational demand and regularity restrictions compared to the conventional phase-field LB models. The first model represents a sharp interface one-fluid formulation, where the LB equation is assigned to solve for a single virtual fluid and the interface is captured through convection of an initially signed distance level set function governed by the level set equation (LSE). The second scheme proposes a diffuse pressure evolution description of the LBE, solving for velocity and dynamic pressure only. In contrast to the common kineticbased solutions of the Cahn-Hilliard equations, the density is then solved via a mass conserving LS equation which benefits from a fast monolithic reinitialization. Rigorous comparisons against established numerical solutions of multiphase NS equations for rising bubble problems are carried out in two and three dimensions, which further provide an unprecedented basis to evaluate the effect of different numerical and implementation aspects of the schemes on the overall performance and accuracy. The simulations are eventually applied to other physically interesting multiphase problems, featuring the flexibility and stability of the scheme under high Re numbers and very large deformations. On the computational side, an efficient implementation of the proposed schemes is presented in particular for manycore general purpose graphical processing units (GPGPU). Various segments of the solution algorithm are then evaluated with respect to their corresponding computational workload and efficient implementation outlines are addressed

Numerical solution of the 2+1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays

In this work we present a formulation of the Teukolsky equation for generic spin perturbations on the hyperboloidal and horizon penetrating foliation of Kerr recently proposed by Racz and Toth. An additional, spin-dependent rescaling of the field variable can be used to achieve stable, long-term, and accurate time-domain evolutions of generic spin perturbations. As an application (and a severe numerical test), we investigate the late-time decays of electromagnetic and gravitational perturbations at the horizon and future null infinity by means of 2+1 evolutions. As initial data we consider four combinations of (non-)stationary and (non-)compact-support initial data with a pure spin-weighted spherical harmonic profile. We present an extensive study of late time decays of axisymmetric perturbations. We verify the power-law decay rates predicted analytically, together with a certain "splitting" behaviour of the power-law exponent. We also present results for non-axisymmetric perturbations. In particular, our approach allows to study the behaviour of the late time decays of gravitational fields for nearly extremal and extremal black holes. For rapid rotation we observe a very prolonged, weakly damped, quasi-normal-mode phase. For extremal rotation the field at future null infinity shows an oscillatory behaviour decaying as the inverse power of time, while at the horizon it is amplified by several orders of magnitude over long time scales. This behaviour can be understood in terms of the superradiance cavity argument

Simulation of forced deformable bodies interacting with two-dimensional incompressible flows: Application to fish-like swimming

International audienceWe present an efficient algorithm for simulation of deformable bodies interacting with two-dimensional incompressible flows. The temporal and spatial discretizations of the Navier-Stokes equations in vorticity stream-function formulation are based on classical fourth-order Runge-Kutta and compact finite differences, respectively. Using a uniform Cartesian grid we benefit from the advantage of a new fourth-order direct solver for the Poisson equation to ensure the incompressibility constraint down to machine zero. For introducing a deformable body in fluid flow, the volume penalization method is used. A Lagrangian structured grid with prescribed motion covers the deformable body interacting with the surrounding fluid due to the hydrodynamic forces and moment calculated on the Eulerian reference grid. An efficient law for curvature control of an anguilliform fish, swimming to a prescribed goal, is proposed. Validation of the developed method shows the efficiency and expected accuracy of the algorithm for fish-like swimming and also for a variety of fluid/solid interaction problems