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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.
- 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
- 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
- ItemNumerical algorithms for Schrödinger equation with artificial boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Čiegis, Raimondas; Laukaitytė, Inga; Radziunas, MindaugasWe consider a one-dimensional linear Schrödinger problem defined on an infinite domain and approximated by the Crank-Nicolson type finite difference scheme. To solve this problem numerically we restrict the computational domain by introducing the reflective, absorbing or transparent artificial boundary conditions. We investigate the conservativity of the discrete scheme with respect to the mass and energy of the solution. Results of computational experiments are presented and the efficiency of different artificial boundary conditions is discussed.
- ItemConstrained Delaunay tetrahedral mesh generation and refinement(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hang, SiA it constrained Delaunay tetrahedralization of a domain in $mathbbR^3$ is a tetrahedralization such that it respects the boundaries of this domain, and it has properties similar to those of a Delaunay tetrahedralization. Such objects have various applications such as finite element analysis, computer graphics rendering, geometric modeling, and shape analysis. This article is devoted to presenting recent developments on constrained Delaunay tetrahedralizations of piecewise linear domains. The focus is for the application of numerically solving partial differential equations using finite element or finite volume methods. We survey various related results and detail two core algorithms that have provable guarantees and are amenable to practical implementation. We end this article by listing a set of open questions.
- ItemHölder index for density states of (α,1,β)-superprocesses at a given point(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Fleischmann, Klaus; Mytnik, Leonid; Wachtel, VitaliA Hölder regularity index at given points for density states of (alpha,1,beta)-superprocesses with alpha>1+beta is determined. It is shown that this index is strictly greater than the optimal index of local Holder continuity for those density states.
- ItemStochastic flow simulation and particle transport in a 2D layer of random porous medium(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Kurbanmuradov, Orazgeldi A.; Sabelfeld, Karl K.A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.
- ItemThe Airy1 process is not the limit of the largest eigenvalue in GOE matrix diffusion(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Bornemann, Folkmar; Ferrari, Patrik; Prähofer, MichaelUsing a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1-process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion
- ItemRandom walks conditioned to stay in Weyl chambers of type C and D(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) König, Wolfgang; Schmid, PatrickWe construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob $h$-transform. Furthermore, we prove functional limit theorems for the rescaled random walks. This is an extension of recent work by Eichelsbacher and König who studied the analogous conditioning for the Weyl chamber of type A. Our proof follows recent work by Denisov and Wachtel who used martingale properties and a strong approximation of random walks by Brownian motion. Therefore, we are able to keep minimal moment assumptions. Finally, we present an alternate function that is amenable to an $h$-transform in the Weyl chamber of type C.