Comparison of three spectral methods for the Benjamin-Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions |
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Authors: | John P. Boyd Zhengjie Xu |
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Affiliation: | a Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor MI 48109-2143, United Statesb Program in Applied and Interdisciplinary Mathematics, University of Michigan, East Hall, Ann Arbor MI 48109, United States |
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Abstract: | The Benjamin-Ono equation is especially challenging for numerical methods because (i) it contains the Hilbert transform, a nonlocal integral operator, and (ii) its solitary waves decay only as O(1/|x|2). We compare three different spectral methods for solving this one-space-dimensional equation. The Fourier pseudospectral method is very fast through use of the Fast Fourier Transform (FFT), but requires domain truncation: replacement of the infinite interval by a large but finite domain. Such truncation is unnecessary for a rational basis, but it is simple to evaluate the Hilbert Transform only when the usual rational Chebyshev functions TBn(x) are replaced by their cousins, the Christov functions; the FFT still applies. Radial basis functions (RBFs) are slow for a given number of grid points N because of the absence of a summation algorithm as fast as the FFT; because RBFs are meshless, however, very flexible grid adaptation is possible. |
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Keywords: | Benjamin-Ono equation Pseudospectral method Rational Chebyshev functions Christov functions Radial basis functions |
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