Optimal Spectral Schemes Based on Generalized Prolate Spheroidal Wave Functions of Order $$$$ |
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Authors: | Jing Zhang Li-Lian Wang Huiyuan Li Zhimin Zhang |
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Affiliation: | 1.School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences,Central China Normal University,Wuhan,China;2.Division of Mathematical Sciences, School of Physical and Mathematical Sciences,Nanyang Technological University,Singapore,Singapore;3.State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software,Chinese Academy of Sciences,Beijing,China;4.Beijing Computational Sciences and Research Center,Beijing,China;5.Department of Mathematics,Wayne State University,Detroit,USA |
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Abstract: | We introduce a family of generalized prolate spheroidal wave functions (PSWFs) of order \(-1,\) and develop new spectral schemes for second-order boundary value problems. Our technique differs from the differentiation approach based on PSWFs of order zero in Kong and Rokhlin (Appl Comput Harmon Anal 33(2):226–260, 2012); in particular, our orthogonal basis can naturally include homogeneous boundary conditions without the re-orthogonalization of Kong and Rokhlin (2012). More notably, it leads to diagonal systems or direct “explicit” solutions to 1D Helmholtz problems in various situations. Using a rule optimally pairing the bandwidth parameter and the number of basis functions as in Kong and Rokhlin (2012), we demonstrate that the new method significantly outperforms the Legendre spectral method in approximating highly oscillatory solutions. We also conduct a rigorous error analysis of this new scheme. The idea and analysis can be extended to generalized PSWFs of negative integer order for higher-order boundary value and eigenvalue problems. |
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