TY - GEN A1 - Yevick, David A1 - Friese, Tilmann A1 - Schmidt, Frank T1 - A Comparison of Transparent Boundary Conditions for the Fresnel Equation N2 - We establish the relationship between the transparent boundary condition (BPP) of Baskakov and Popov [Wave Motion 14 (1991) 121-128] and Pakpadakis et. al. [J. Acoust. Soc. Am. 92 (1992) 2030-2038] and a second boundary condition (SDY) introduced by Schmidt and Deuflhard [Comp. Math. Appl. 29 (1995) 53-76] and Schmidt and Yevick [J. Compu. Phys. 134 (1997) 96-107], that is explicitly tailored to the form of the underlying numerical propagation scheme. Our analysis demonstrates that if the domain is first discretized in the propagation direction, the SDY expression can be obtained by applying the exact sequence of steps used to derive the BPP procedure. The BPP method is thus an approximate realization of the computationally far simpler and unconditionally stable SDY boundary condition. T3 - ZIB-Report - 00-05 KW - Fresnel equation KW - transparent boundary condition Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5738 ER - TY - GEN A1 - Friese, Tilman A1 - Schmidt, Frank A1 - Yevick, David T1 - Transparent Boundary Conditions for a Wide-Angle Approximation of the One-Way Helmholtz Equation N2 - We present nonlocal discrete transparent boundary conditions for a fourth-order wide-angle approximation of the two-dimensional Helmholtz equation. The boundary conditions are exact in the sense that they supply the same discrete solution on a bounded interior domain as would be obtained by considering the problem on the entire unbounded domain with zero boundary conditions at infinity. The proposed algorithm results in an unconditionally stable propagation method. Numerical examples from optics illustrate the efficiency of our approach. T3 - ZIB-Report - SC-99-45 KW - Helmholtz equation KW - Wide-angle approximation KW - Transparent boundary conditions KW - Finite-element method Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4345 ER - TY - GEN A1 - Schmidt, Frank A1 - Friese, Tilmann A1 - Yevick, David T1 - Transparent Boundary Conditions for Split-Step Pade Approximations of the One-Way Helmholtz Equation N2 - In this paper, we generalize the nonlocal discrete transparent boundary condition introduced by Schmidt and Deuflhard {[}Comp. Math. Appl. 29 (1995) 53-76{]} and Schmidt and Yevick {[}J. Comput. Phys. 134 (1997) 96-107{]} to propagation methods based on arbitrary Pad\'e approximations to the two-dimensional one-way Helmholtz equation. Our approach leads to a recursive formula for the coefficients appearing in the nonlocal condition which then yields an unconditionally stable propagation method. T3 - ZIB-Report - SC-99-46 KW - Helmholtz equation KW - Wide-angle approximation KW - Pad/'e split step approximation KW - Transparent boundary conditions KW - Finite-element method Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4353 ER - TY - GEN A1 - Schmidt, Frank A1 - Yevick, David T1 - Discrete Transparent Boundary Conditions for Schrödinger-Type Equations N2 - We present a general technique for constructing nonlocal transparent boundary conditions for one-dimensional Schrödinger-type equations. Our method supplies boundary conditions for the $\theta$-family of implicit one-step discretizations of Schrödinger's equation in time. The use of Mikusi\'nski's operator approach in time avoids direct and inverse transforms between time and frequency domains and thus implements the boundary conditions in a direct manner. T3 - ZIB-Report - SC-96-17 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2289 ER - TY - JOUR A1 - Schmidt, Frank A1 - Friese, T. A1 - Yevick, David T1 - Transparent Boundary Conditions for Split-Step Padé Approximations of the One-Way Helmholtz equation JF - J. Comp. Phys Y1 - 2001 VL - 170 SP - 696 EP - 719 ER - TY - JOUR A1 - Friese, T. A1 - Schmidt, Frank A1 - Yevick, David T1 - Transparent boundary conditions for a wide-angle approximation of the one-way Helmholtz equation JF - J. Comput. Phys. Y1 - 2000 U6 - https://doi.org/10.1006/jcph.2000.6637 VL - 165 IS - 2 SP - 645 EP - 659 ER - TY - JOUR A1 - Yevick, David A1 - Friese, T. A1 - Schmidt, Frank T1 - A Comparison of Transparent Boundary Conditions for the Fresnel Equation JF - J. Comput. Phys. Y1 - 2001 VL - 168 IS - 2 SP - 433 EP - 444 ER - TY - JOUR A1 - Yevick, David A1 - Yu, J. A1 - Schmidt, Frank T1 - Analytic Studies of Absorbing and Impedance-Matched Boundary Layers JF - IEEE Photonics Technology Letters Y1 - 1997 U6 - https://doi.org/10.1109/68.554499 VL - 9 SP - 73 EP - 75 ER - TY - JOUR A1 - Schmidt, Frank A1 - Yevick, David T1 - Discrete Transparent Boundary Conditions for Schrödinger-Type Equations JF - J. Comput. Phys. Y1 - 1997 U6 - https://doi.org/10.1006/jcph.1997.5675 VL - 134 IS - 1 SP - 96 EP - 107 ER - TY - CHAP A1 - Schmidt, Frank A1 - Yevick, David T1 - Discrete Transparent Boundary Conditions for the Fresnel Equation T2 - Proceedings of the 8th European Conference on Integrated Optics Y1 - 1997 SP - 222 EP - 225 CY - Royal Institute of Technology, Stockholm, Sweden ER -