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

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.

[1]  Jerry L. Bona,et al.  SINGULARITY FORMATION IN THE GENERALIZED BENJAMIN-ONO EQUATION , 2004 .

[2]  John P. Boyd.  The orthogonal rational functions of Higgins and Christov and algebraically mapped Chebyshev polynomials , 1990 .

[3]  C. Christov.  A Complete Orthonormal System of Functions in $L^2 ( - \infty ,\infty )$ Space , 1982 .

[4]  J. Boyd.  Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics , 1998 .

[5]  P. Miller,et al.  The Benjamin-Ono Hierarchy with Asymptotically Reflectionless Initial Data in the Zero-Dispersion Limit , 2011, 1101.0784.

[6]  Beatrice Pelloni,et al.  Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations , 2001 .

[7]  Heping Ma,et al.  Optimal error estimates of the Fourier spectral method for a class of nonlocal, nonlinear dispersive wave equations , 2009 .

[8]  T. Miloh,et al.  A note on the numerical and N-soliton solutions of the Benjamin-Ono evolution equation , 1993 .

[9]  Heping,et al.  Error Estimate of the Fourier Collocation Method for the Benjamin-Ono Equation , 2009 .

[10]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[11]  Christo I. Christov,et al.  Fourier-Galerkin method for 2D solitons of Boussinesq equation , 2007, Math. Comput. Simul..

[12]  Zhen-hui Xu.  Asymptotic Analysis and Numerical Analysis of the Benjamin-Ono Equation , 2010 .

[13]  H. Kalisch.  Error Analysis of a Spectral Projection of the Regularized Benjamin–Ono Equation , 2005 .

[14]  P. Miller,et al.  On the zero‐dispersion limit of the benjamin‐ono cauchy problem for positive initial data , 2010, 1002.3178.

[15]  David J. Kaup,et al.  The Inverse Scattering Transform for the Benjamin–Ono Equation , 1998 .

[16]  John P. Boyd.  Evaluating of Dawson's Integral by solving its differential equation using orthogonal rational Chebyshev functions , 2008, Appl. Math. Comput..

[17]  J. R. Higgins.  Completeness and basis properties of sets of special functions: Bibliography , 1977 .

[18]  John P. Boyd,et al.  The uselessness of the Fast Gauss Transform for summing Gaussian radial basis function series , 2010, J. Comput. Phys..

[19]  J. Weideman.  Computations of the complex error function , 1994 .

[20]  J. Boyd.  Spectral methods using rational basis functions on an infinite interval , 1987 .

[21]  J. A. C. Weideman,et al.  Computing the Hilbert transform on the real line , 1995 .

[22]  V. Thomée,et al.  A numerical method for the Benjamin-Ono equation , 1998 .