TY - GEN A1 - Hohage, Thorsten A1 - Schmidt, Frank T1 - On the Numerical Solution of Nonlinear Schrödinger type equations in fiber optics N2 - The aim of this paper is to develop fast methods for the solution of nonlinear Schrödinger type equations in fiber optics. Using the method of lines we have to solve a stiff system of ordinary differential equations where the eigenvalues of the Jacobian are close to the imaginary axis. This is usually done by a Split Step method. Here we consider the extrapolation of Split Step methods with adaptive order and step size control. For more complicated nonlinearities, in particular stimulated Raman scattering, Split Step methods are less efficient since symmetry is either destroyed or requires much additional effort. In this case we use implicit Runge Kutta formulas of Gauß type. The key point for the efficient implementation of these methods is that the system of nonlinear algebraic equations can be solved without setting up the Jacobian. The proposed methods are compared to other methods, in particular exponential integrators, the method of Marcuse, and the method of Blow and Wood. T3 - ZIB-Report - 02-04 KW - nonlinear Schroedinger equation KW - split step methods KW - extrapolation KW - collocation methods Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6715 ER - TY - GEN A1 - Hohage, Thorsten A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Solving time-harmonic scattering problems based on the pole condition: Convergence of the PML method N2 - In this paper we study the PML method for Helmholtz-type scattering problems with radially symmetric potential. The PML method consists in surrounding the computational domain by a \textbf{P}erfectly \textbf{M}atched sponge \textbf{L}ayer. We prove that the approximate solution obtained by the PML method converges exponentially fast to the true solution in the computational domain as the thickness of the sponge layer tends to infinity. This is a generalization of results by Lassas and Somersalo based on boundary integral eqaution techniques. Here we use techniques based on the pole condition instead. This makes it possible to treat problems without an explicitly known fundamental solution. T3 - ZIB-Report - 01-23 KW - transparent boundary conditions KW - PML KW - pole condition Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6494 ER - TY - GEN A1 - Hohage, Thorsten A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - A new method for the solution of scattering problems N2 - We present a new efficient algorithm for the solution of direct time-harmonic scattering problems based on the Laplace transform. This method does not rely on an explicit knowledge of a Green function or a series representation of the solution, and it can be used for the solution of problems with radially symmetric potentials and problems with waveguides. The starting point is an alternative characterization of outgoing waves called \emph{pole condition}, which is equivalent to Sommerfeld's radiation condition for problems with radially symmetric potentials. We obtain a new representation formula, which can be used for a numerical evaluation of the exterior field in a postprocessing step. Based on previous theoretical studies, we discuss the numerical realization of our algorithm and compare its performance to the PML method. T3 - ZIB-Report - 02-01 KW - transparent boundary conditions KW - inhomogeneous exterior domains KW - Laplace transform Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6683 ER - TY - GEN A1 - Hohage, Thorsten A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Solving time-harmonic scattering problems based on the condition: Theory N2 - The pole condition is a general concept for the theoretical analysis and the numerical solution of a variety of wave propagation problems. It says that the Laplace transform of the physical solution in radial direction has no poles in the lower complex half-plane. In the present paper we show that for the Helmholtz equation with a radially symmetric potential the pole condition is equivalent to Sommerfeld's radiation condition. Moreover, a new representation formula based on the pole condition is derived and used to prove existence, uniqueness and asymptotic properties of solutions. This lays the foundations of a promising new algorithm to solve time-harmonic scattering problems numerically and provides a new approach for analyzing existing algorithms such as the Perfectly Matched Layer (PML) method and the Bayliss-Gunzburger-Turkel (BGT) algorithm. T3 - ZIB-Report - 01-01 KW - transparent boundary conditions KW - Laplace transform KW - Sommerfeld radiation condition Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6279 ER - TY - JOUR A1 - Schmidt, Frank A1 - Hohage, Thorsten A1 - Klose, Roland A1 - Schädle, Achim A1 - Zschiedrich, Lin T1 - Pole condition: A numerical method for Helmholtz-type scattering problems with inhomogeneous exterior domain JF - J. Comput. Appl. Math. Y1 - 2008 U6 - https://doi.org/10.1016/j.cam.2007.04.046 VL - 218 IS - 1 SP - 61 EP - 69 ER - TY - CHAP A1 - Hohage, Thorsten A1 - Schmidt, Frank A1 - Zschiedrich, Lin ED - Michielsen, Bas ED - Decavèle, Francine T1 - A new method for the solution of scattering problems T2 - Proceedings of the JEE’02 Symposium Y1 - 2002 SP - 251 EP - 256 PB - ONERA CY - Toulouse ER - TY - JOUR A1 - Hohage, Thorsten A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Solving Time-Harmonic Scattering Problems Based on the Pole Condition I: Theory JF - SIAM J. Math. Anal. Y1 - 2003 U6 - https://doi.org/10.1137/S0036141002406473 VL - 35 IS - 1 SP - 183 EP - 210 ER - TY - JOUR A1 - Hohage, Thorsten A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Solving Time-Harmonic Scattering Problems Based on the Pole Condition II: Convergence of the PML Method JF - SIAM J. Math. Anal. Y1 - 2003 U6 - https://doi.org/10.1137/S0036141002406485 VL - 35 IS - 3 SP - 547 EP - 560 ER -