Comparison of three spectral methods for the Benjamin–Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions

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.

Research Highlights
► We compare three exponentially convergent algorithms for the Benjamin–Ono equation.
► The BO Eq. is challenging because of non-local integral operator and slowly decaying solitons.
► Christov functions are eigenfunctions of Hilbert integral; preferred rational basis.
► Fourier and Christov basis are fast because the Fast Fourier Transform applies.
► Radial basis functions are slow, but allow adaptivity without coordinate mapping.