Fast eighth-order Padé schemes based on Chebyshev polynomials for the direct Zakharov-Shabat problem
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Sergey Medvedev1,2, Igor Chekhovskoy1,2, Irina Vaseva1,2
1Federal Research Center for Information and Computational Technologies, Novosibirsk 630090, Russia.
Abstract:
In this work, we construct fast eighth-order Padé schemes for the direct Zakharov-Shabat scattering problem. The schemes are based on an eighth-order exponential integrator obtained from the Magnus expansion. A direct extension of the conventional fast representation to the eighth-order case leads to insufficiently accurate results in the considered tests. To overcome this difficulty, we reformulate the spectral dependence through the Joukowski mapping and the Chebyshev polynomial basis. Numerical experiments with chirped hyperbolic secant potentials show improved accuracy of the Chebyshev representation in resolved regimes. Its accuracy and computational cost are compared with direct-Cayley fast schemes for different spectral intervals and scaling parameters.
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