@misc{HohageSchmidt, author = {Hohage, Thorsten and Schmidt, Frank}, title = {On the Numerical Solution of Nonlinear Schr{\"o}dinger type equations in fiber optics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6715}, number = {02-04}, abstract = {The aim of this paper is to develop fast methods for the solution of nonlinear Schr{\"o}dinger 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.}, language = {en} } @misc{HohageSchmidtZschiedrich, author = {Hohage, Thorsten and Schmidt, Frank and Zschiedrich, Lin}, title = {Solving time-harmonic scattering problems based on the pole condition: Convergence of the PML method}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6494}, number = {01-23}, abstract = {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.}, language = {en} } @misc{HohageSchmidtZschiedrich, author = {Hohage, Thorsten and Schmidt, Frank and Zschiedrich, Lin}, title = {A new method for the solution of scattering problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6683}, number = {02-01}, abstract = {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.}, language = {en} } @misc{HohageSchmidtZschiedrich, author = {Hohage, Thorsten and Schmidt, Frank and Zschiedrich, Lin}, title = {Solving time-harmonic scattering problems based on the condition: Theory}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6279}, number = {01-01}, abstract = {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.}, language = {en} } @article{SchmidtHohageKloseetal.2008, author = {Schmidt, Frank and Hohage, Thorsten and Klose, Roland and Sch{\"a}dle, Achim and Zschiedrich, Lin}, title = {Pole condition: A numerical method for Helmholtz-type scattering problems with inhomogeneous exterior domain}, series = {J. Comput. Appl. Math.}, volume = {218}, journal = {J. Comput. Appl. Math.}, number = {1}, doi = {10.1016/j.cam.2007.04.046}, pages = {61 -- 69}, year = {2008}, language = {en} } @inproceedings{HohageSchmidtZschiedrich2002, author = {Hohage, Thorsten and Schmidt, Frank and Zschiedrich, Lin}, title = {A new method for the solution of scattering problems}, series = {Proceedings of the JEE'02 Symposium}, booktitle = {Proceedings of the JEE'02 Symposium}, editor = {Michielsen, Bas and Decav{\`e}le, Francine}, publisher = {ONERA}, address = {Toulouse}, pages = {251 -- 256}, year = {2002}, language = {en} } @article{HohageSchmidtZschiedrich2003, author = {Hohage, Thorsten and Schmidt, Frank and Zschiedrich, Lin}, title = {Solving Time-Harmonic Scattering Problems Based on the Pole Condition I: Theory}, series = {SIAM J. Math. Anal.}, volume = {35}, journal = {SIAM J. Math. Anal.}, number = {1}, doi = {10.1137/S0036141002406473}, pages = {183 -- 210}, year = {2003}, language = {en} } @article{HohageSchmidtZschiedrich2003, author = {Hohage, Thorsten and Schmidt, Frank and Zschiedrich, Lin}, title = {Solving Time-Harmonic Scattering Problems Based on the Pole Condition II: Convergence of the PML Method}, series = {SIAM J. Math. Anal.}, volume = {35}, journal = {SIAM J. Math. Anal.}, number = {3}, doi = {10.1137/S0036141002406485}, pages = {547 -- 560}, year = {2003}, language = {en} }