Commun. Comput. Phys., 5 (2009), pp. 849-870. |
Numerical Simulation of Waves in Periodic Structures Matthias Ehrhardt ^{1}, Houde Han ^{2}, Chunxiong Zheng ^{2*} 1 Weierstras-Institut fur Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany.2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China. Received 28 February 2008; Accepted (in revised version) 26 June 2008 Available online 29 September 2008 Abstract In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu [J. Lightwave Technology 25 (2007), 3649-3656] for solving periodic array problems. It turns out that when the periodic array contains an infinite number of periodic cells, our method gives a fast evaluation of the exact boundary Robin-to-Robin mapping if the wave number is complex, or real but in the stop bands. This technique is also used to solve the time-dependent Schrodinger equation in both one and two dimensions, when the periodic potential functions have some local defects. AMS subject classifications: 35B27, 65M99, 35Q60, 35J05PACS: 02.70.Bf, 31.15.-p, 42.82.Et, 85.35.-p, 85.35.Be Key words: Periodic media, Helmholtz equation, Schrodinger equation, Dirichlet-to-Neumann maps, Robin-to-Robin maps, band structure, Floquet-Bloch theory, high-order finite elements. *Corresponding author. Email: ehrhardt@wias-berlin.de (M. Ehrhardt), hhan@math.tsinghua.edu.cn (H. Han), czheng@math.tsinghua.edu.cn (C. Zheng) |