# Preprints aus dem Institut für Mathematik

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Preprints aus dem Institut für Mathematik

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Publication 2013-09-12BuchOperator-Splitting Methods Respecting Eigenvalue Problems For Shallow Shelf Equations With Basal DragCalov, Reinhard; Geiser, JürgenWe discuss different numerical methods for solving the shallow shelf equations with basal drag. The coupled equations are decomposed into operators for membranes stresses, basal shear stress and driving stress. Applying reasonable parameter values, we demonstrate that the operator of the membrane stresses is much stiffer than operator of the basal shear stress. Therefore, we propose a new splitting method, which alternates between the iteration on the membrane-stress operator and the basal-shear operator, with a stronger iteration on the operator of the membrane stress. We show that this splitting improves the computational performance of the numerical method, although the choice of the (standard) method to solve for all operators in one step speeds up the scheme too. (Based on the delicate and coupled equation we propose a new decomposition method to decouple into simpler solvable sub-equations. After a number of approximations we consider the error of the method and proposed a choice of the operators.)Publication 2011-10-20BuchReview of AdS/CFT Integrability, Chapter I.2Sieg, ChristophWe review the constructions and tests of the dilatation operator and of the spectrum of composite operators in the flavour SU(2) subsector of N=4 SYM in the planar limit by explicit Feynman graph calculations with emphasis on analyses beyond one loop. From four loops on, the dilatation operator determines the spectrum only in the asymptotic regime, i.e. to a loop order which is strictly smaller than the number of elementary fields of the composite operators. We review also the calculations which take a first step beyond this limitation by including the leading wrapping corrections.Publication 2011-10-20BuchReview of AdS/CFT IntegrabilityBeisert, Niklas; Ahn, Changrim; Alday, Luis F.; Bajnok, Zoltán; Drummond, James M.; Freyhult, Lisa; Gromov, Nikolay; Janik, Romuald A.; Kazakov, Vladimir; Klose, Thomas; Korchemsky, Gregory P.; Kristjansen, CharlotteThis is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection we present an overview of the achievements and the status of this subject as of the year 2010.Publication 2011-09-27BuchTwist operators in N=4 beta-deformed theoryLeeuw, Marius de; Lukowski, TomaszIn this paper we derive both the leading order finite size corrections for twist-2 and twist-3 operators and the next-to-leading order finite-size correction for twist-2 operators in beta-deformed SYM theory. The obtained results respect the principle of maximum transcendentality as well as reciprocity. We also find that both wrapping corrections go to zero in the large spin limit. Moreover, for twist-2 operators we studied the pole structure and compared it against leading BFKL predictions.Publication 2011-09-27BuchReview of AdS/CFT Integrability, Chapter III.1Staudacher, MatthiasThe one-dimensional Heisenberg XXX spin chain appears in a special limit of the AdS/CFT integrable system. We review various ways of proving its integrability, and discuss the associated methods of solution. In particular, we outline the coordinate and the algebraic Bethe ansatz, giving reference to literature suitable for learning these techniques. Finally we speculate which of the methods might lift to the exact solution of the AdS/CFT system, and sketch a promising method for constructing the Baxter Q-operator of the XXX chain. It allows to find the spectrum of the model using certain algebraic techniques, while entirely avoiding Bethe's ansatz.Publication 2011-09-27BuchFrom convergence principles to stability and optimalityKlatte, Diethard; Kruger, Alexander; Kummer, BerndWe show in a rather general setting that Hoelder and Lipschitz stability properties of solutions to variational problems can be characterized by convergence of more or less abstract iteration schemes. Depending on the principle of convergence, new and intrinsic stability conditions can be derived. Our most abstract models are (multi-) functions on complete metric spaces. The relevance of this approach is illustrated by deriving both classical and new results on existence and optimality conditions, stability of feasible and solution sets and convergence behavior of solution procedures.Publication 2011-09-27BuchIterative Operator Splitting Method for Coupled ProblemsGeiser, Jürgen; Küttel, FelixIn this article a new approach is considered for implementing operator splitting methods for transport problems, influenced by eletric fields. Our motivation came to model PE-CVD (plasma-enhanced chemical vapor deposition) processes, means the flow of species to a gas-phase, which are influenced by an electric field. We consider a convection-diffusion equation and a Lorence force in the electrostatic case. The iterative splitting schemes is given as an embedded coupling method and we apply such a scheme as a fast solver. The decomposition analysis is discussed for the nonlinear case. Numerical experiments are given with respect to explicit Adam-Bashforth schemes. We discuss the convergence behavior in time and space for the iterative schemes.Publication 2011-09-27BuchOperator-Splitting Methods Respecting Eigenvalue Problems For Shallow Shelf Equations With Basal DragGeiser, Jürgen; Calov, Reinhard; Recknagel, ThomasWe discuss different numerical methods for solving the shallow shelf equations with basal drag. The coupled equations are decomposed into operators for membranes stresses, basal shear stress and driving stress. Applying reasonable parameter values, we demonstrate that the operator of the membrane stresses is much stiffer than operator of the basal shear stress. Therefore, we propose a new splitting method, which alternates between the iteration on the membrane-stress operator and the basal-shear operator, with a stronger iteration on the operator of the membrane stress. We show that this splitting improves the computational performance of the numerical method, although the choice of the (standard) method to solve for all operators in one step speeds up the scheme too. (Based on the delicate and coupled equation we propose a new decomposition method to decouple into simpler solvable sub-equations. After a number of approximations we consider the error of the method and proposed a choice of the operators.)Publication 2011-09-27BuchSuperspace calculation of the four-loop spectrum in N=6 supersymmetricChern-Simons theoryLeoni, Matias; Mauri, Andrea; Minahan, Joseph A.; Sax, Olof Ohlsson; Santambrogio, Alberto; Sieg, Christoph; Tartaglino-Mazzucchelli, GabrielePublication 2011-09-27BuchPorous Media Based Modeling of parallel Plate PE-CVD ApparatusGeiser, Jürgen; Arab, M.The numerous technical applications in deposit metal plates with new materials like SiC and TiC has an advantage to overcome the leaking corrosive behavior and have additional a good electrical behavior. Here we present an application of a porous media to model a homogenized deposition with a parallel plate PE-CVD apparatus. Special geometries of parallel Anodes and cathodes helps to obtain at least a laminar flow field. By the way the delicate arrangement of the anode and cathode has to be simulated. The flux of the precursors are important to simulate to the porous media given as the plasma background. Here we can optimize the transport to the delicate geometry respecting the flux field in the permeable layers. To derive a mathematical model, we deal with a model for the transport and kinetics of the different species. Underlying physical experiments help to approximate the parameters of the numerical model. We introduce a multi regression method to approximate the physical to the mathematical parameters. We present results of some numerical simulations and help to foresee some effects to find on optimal deposition process.Publication 2011-09-27BuchIterative Operator Splitting Methods For Differential EquationsGeiser, JürgenIn this paper we describe an iterative operator-splitting method for bounded operators. Our contribution is a novel iterative method that can be applied as a splitting method to ordinary and partial differential equations. A simple relation between the number of iterative steps and order of the splitting scheme makes this an alternative method to a time decomposition method. The iterative splitting scheme can be applied to a physical problem, but the original problem is not divided as in standard splitting schemes. We present error bounds for iterative splitting methods in the presence of bounded operators. We discuss efficient algorithms for computing the integral formulation of the splitting scheme. In experiments, we consider the benefits of the novel splitting method in terms of the number of iterations and time steps. Ordinary differential equations and convection-diffusion-reaction equations are presented in the numerical results.Publication 2011-09-27BuchOscillator Construction of su(n|m) Q-OperatorsFrassek, Rouven; Lukowski, Tomasz; Meneghelli, Carlo; Staudacher, MatthiasWe generalize our recent explicit construction of the full hierarchy of Baxter Q-operators of compact spin chains with su(n) symmetry to the supersymmetric case su(n|m). The method is based on novel degenerate solutions of the graded Yang-Baxter equation, leading to an amalgam of bosonic and fermionic oscillator algebras. Our approach is fully algebraic, and leads to the exact solution of the associated compact spin chains while avoiding Bethe ansatz techniques. It furthermore elucidates the algebraic and combinatorial structures underlying the system of nested Bethe equations. Finally, our construction naturally reproduces the representation, due to Z. Tsuboi, of the hierarchy of Baxter Q-operators in terms of hypercubic Hasse diagrams.Publication 2011-09-27BuchOn the existence of weak solutions to a coupled system of two turbulent flowsNeumann, Joachim; Wolf, JörgPublication 2011-09-27BuchAlgorithms of intrinsic complexity for point searching in real singular hypersurfacesBank, Bernd; Giusti, Marc; Heintz, Joos; Lehmann, Lutz; Pardo, Luis-MiguelPublication 2011-09-27BuchMagnus integrator and successive approximation for solving time-dependent problemsGeiser, JürgenMagnus integrator and successive approximation for solving time-dependent problems. The Magnus expansion has been intensely studied and widely applied for solving explicitly time-dependent problems. Due to its exponential character, it is rather difficult to derive practical algorithms beyond the sixth-order. An alternative method is based on successive approximation methods, that taken into account the temporally inhomogeneous equation (method of Tanabe and Sobolevski). In this work, we show that the recently derived ideas of the successive approximation method in a splitting method. Examples are discussed.Publication 2011-09-27BuchA Shortcut to the Q-OperatorBazhanov, Vladimir V.; Lukowski, Tomasz; Meneghelli, Carlo; Staudacher, MatthiasBaxter's Q-operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin-1/2 Heisenberg-Bethe XXX spin chain. Here we attempt to fill this gap and show how two linearly independent operatorial solutions to Baxter's TQ equation may be constructed as commuting transfer matrices if a twist field is present. The latter are obtained by tracing over infinitely many oscillator states living in the auxiliary channel of an associated monodromy matrix. We furthermore compare and differentiate our approach to earlier articles addressing the problem of the construction of the Q-operator for the XXX chain. Finally we speculate on the importance of Q-operators for the physical interpretation of recent proposals for the Y-system of AdS/CFT.Publication 2011-09-27BuchBaxter Q-Operators and Representations of YangiansBazhanov, Vladimir V.; Frassek, Rouven; Lukowski, Tomasz; Meneghelli, Carlo; Staudacher, MatthiasWe develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the Yang-Baxter equation connected with harmonic oscillator algebras. These infinite-state solutions of the Yang-Baxter equation serve as elementary, "partonic" building blocks for other solutions via the standard fusion procedure. As a first example of the method we consider sl(n) compact spin chains and derive the full hierarchy of operatorial functional equations for all related commuting transfer matrices and Q-operators. This leads to a systematic and transparent solution of these chains, where the nested Bethe equations are derived in an entirely algebraic fashion, without any reference to the traditional Bethe ansatz techniques.Publication 2011-09-27BuchOn the construction of a class of Dulac-Cherkas functions for generalized Liénard systemsCherkas, Leonid; Grin, Alexander; Schneider, KlausPublication 2011-09-27BuchMonte Carlo simulations concerning modeling DC and high power pulsed magnetron sputtering for Ti₃SiC₂ including high pressures and ion deposition probabilitiesGeiser, JürgenWe motivate our study by simulating the particle transport of a thin film deposition process done by PVD (physical vapor deposition) processes. In this paper we present a new model taken into account a higher pressure regimes in a sputter process. We propose a collision models for projectile and target collisions in order to compute the mean free path and include the virial coefficients that considered interacting gas particles. A detailed description of collision models of the Monte Carlo Simulations is discussed for high power impulse magnetron sputtering (HIPIMS) and DC sputtering in lower pressure regimes. We derive an equation for the mean free path for arbitrary interactions (cross sections) which (most important) includes the relative velocity between the projectiles and targets based on physical first principles and extend with higher order Virial terms . At the substrate we simulate the implantation of the particles with the help of TRIM, based on result of the energy that are computed with the Monte Carlo methods. We apply our results to three interaction models: hard sphere interaction, Screened Coulomb interaction and a mixture of the last mentioned interactions. The deposition to realistic geometries, which have sharp angles included, are presented. Because of the strong convective process of a HIPIMS method, the low diffusion process allows not to deposit into delicate geometries, see [Christ2005]. This can be improved by rotating the target to a more or less perpendicular angle.Publication 2011-09-27BuchModel of PE-CVD apparatusGeiser, Jürgen; Buck, V.; Arab, M.In this paper we present the simulation of a chemical vapor deposition for metallic bipolar plates. For chemical vapor deposition, the delicate optimization between temperature, pressure and plasma power is important to obtain a homogeneous depositio. The aim is to reduce real-life experiments of a given CVD plasma reactor, based on a large physical parameter space we have a hugh amount of experiments. A detail study of the physical experiments on a CVD plasma reactor allows to reduce to an approximated mathematical model, which is the underlying transport-reaction model. Significant region of the CVD apparatus are approximated and physical parameters are transferred to the mathematical parameters. Such approximation reduced the mathematical parameter space to a realistic amount of numerical experiments. Based on interpolation and regression functions we fit to the physical parameter space and can give first prediction to deposition rates with the simulation model. Here numerical experiments help to understand the deposition process and the control the positions of the sources for the deposition and precursor gases. For the simulations we apply analytical as well as numerical methods to obtain results to predict the growth of thin layers. The results are discussed with physical experiments to give a valid model for the assumed growth. Here an important transfer of engineering research on modelling real-life processes to achieve a simulatable mathematical model. Such a model can be solved by numerical solvers and discretisation schemes. The results can be used to obtain a new understanding of the technical processes in engineering research.