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- ItemSimulations of 3D/4D precipitation processes in a turbulent flow field(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) John, Volker; Roland, MichaelPrecipitation processes are modeled by population balance systems. A very expensive part of the simulation of population balance systems is the solution of the equation for the particle size distribution (PSD) since this equation is defined in a higher dimensional domain than the other equations in the system. This paper studies different approaches for the solution of this equation: two finite difference upwind schemes and a linear finite element flux--corrected transport method. It is shown that the different schemes lead to qualitatively different solutions for an output of interest.
- ItemPlasma induced pulse breaking in filamentary self-compression(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Brée, Carsten; Demircan, Ayhan; Skupin, Stefan; Berg´e, Luc; Steinmeyer, GünterA plasma induced temporal break-up in filamentary propagation has recently been identified as one of the key events in the temporal self-compression of femtosecond laser pulses. An analysis of the Nonlinear Schrödinger Equation coupled to a noninstantaneous plasma response yields a set of stationary states. This analysis clearly indicates that the emergence of double-hump, characteristically asymmetric temporal on-axis intensity profiles in regimes where plasma defocusing saturates the optical collapse caused by Kerr self-focusing is an inherent property of the underlying dynamical model.
- ItemLinear stability analysis of ta sharp-interface model for dewetting thin films(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) King, John R.; Münch, Andreas; Wagner, BarbaraThe topic of this study concerns the stability of the three-phase contact-line of a dewetting thin liquid film on a hydrophobised substrate driven by van der Waals forces. The role of slippage in the emerging instability at the three-phase contact-line is studied by deriving a sharp-interface model for the dewetting thin film via matched asymptotic expansions. This allows for a derivation of travelling waves and their linear stability via eigenmode analysis. In contrast to the dispersion relations typically encountered for the finger-instabilty, where the dependence of the growth rate on the wave number is quadratic, here it is linear. Using the separation of time scales of the slowly growing rim of the dewetting film and time scale on which the contact line destabilises, the sharp-interface results are compared to earlier results for the full lubrication model and good agreement for the most unstable modes is obtained.
- ItemMaximal convergence theorems for functions of squared modulus holomorphic type and various applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Kraus, ChristianeIn this paper we extend the theory of maximal convergence introduced by Walsh to functions of squared modulus holomorphic type. We introduce in accordance to the well-known complex maximal convergence number for holomorphic functions a real maximal convergence number for functions of squared modulus holomorphic type and prove several maximal convergence theorems. We achieve that the real maximal convergence number for F is always greater or equal than the complex maximal convergence number for g and equality occurs if L is a closed disk in R^2. Among other various applications of the resulting approximation estimates we show that for functions F of squared holomorphic type which have no zeros in a closed disk B_r the relation limsupntoinftysqrt[n]En(Br,F)=limsupntoinftysqrt[n]En(partialBr,F) is valid, where E_n is the polynomial approximation error.
- ItemMathematical results on existence for viscoelastodynamic problems with unilateral constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Petrov, Adrien; Schatzman, M.We study a damped wave equation and the evolution of a Kelvin-Voigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution.
- ItemFast, stable and accurate method for the Black-Scholes equation of American options(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Mickens, Ronald E.We propose a simple model for the behaviour of long-time investors on stock markets, consisting of three particles, which represent the current price of the stock, and the opinion of the buyers, or sellers resp., about the right trading price. As time evolves both groups of traders update their opinions with respect to the current price. The update speed is controled by a parameter $\gamma$, the price process is described by a geometric Brownian motion. The stability of the market is governed by the difference of the buyers' opinion and the sellers' opinion. We prove that the distance
- ItemA mathematical model for case hardening of steel(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Fasano, Antonio; Hömberg, Dietmar; Panizzi, LuciaA mathematical model for the gas carburizing of steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to render the steel austenitic, followed by quenching to form a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the phase fractions. We prove existence and uniqueness of a solution and finally present some numerical simulations.
- ItemRegularity of elastic fields in composites(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Knees, Dorothee; Sändig, Anna-MargareteIt is well known that high stress concentrations can occur in elastic composites in particular due to the interaction of geometrical singularities like corners, edges and cracks and structural singularities like jumping material parameters. In the project C5 "Stress concentrations in heterogeneous materials" of the SFB 404 "Multifield Problems in Solid and Fluid Mechanics" it was mathematically analyzed where and which kind of stress singularities in coupled linear and nonlinear elastic structures occur. In the linear case asymptotic expansions near the geometrical and structural peculiarities are derived, formulae for generalized stress intensity factors included. In the nonlinear case such expansions are unknown in general and regularity results are proved for elastic materials with power-law constitutive equations with the help of the difference quotient technique combined with a quasi-monotone covering condition for the subdomains and the energy densities. Furthermore, some applications of the regularity results to shape and structure optimization and the Griffith fracture criterion in linear and nonlinear elastic structures are discussed. Numerical examples illustrate the results.
- ItemImplementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh-Weyl theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Ehrhardt, Matthias; Zheng, ChunxiongA new approach for simulating the solution of the time-dependent Schrödinger equation with a general variable potential will be proposed. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to a suitable spectral parameter. With the proposed method we can overcome the usual high-frequency restriction for absorbing boundary conditions of general variable potential problems. We end up with a fast computational algorithm for absorbing boundary conditions that are accurate for the full frequency band
- ItemModerate deviations for random walk in random scenery(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Fleischmann, Klaus; Mörters, Pete; Wachtel, VitaliWe investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér's condition. We prove moderate deviation principles in dimensions d ≥ 2, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case d ≥ 4 we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In d ≥ 3, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in $ = 2 we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.