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IMA Journal of Numerical Analysis 2001 21(1):239-263; doi:10.1093/imanum/21.1.239
© 2001 by Institute of Mathematics and its Applications
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Spectral analysis and computation for the Kuramoto–Sakaguchi integroparabolic equation

Juan A. Acebrón1, Mikhail M. Lavrentiev, Jr2 and Renato Spigler3

1 Escuela Politécnica Superior, Universidad Carlos III de Madrid, Spain 2 Sobolev Institute of Mathematics, Siberian Division of the Russian Academy of Sciences, Novosibirsk, Russia 3 Dipartimento di Matematica, Università di ‘Roma Tre’, Rome, Italy

A spectral method is developed to numerically solve the so-called Kuramoto–Sakaguchi equation, which is a nonlinear integro-differential equation of the parabolic type, governing the dynamical statistical behaviour of certain populations of nonlinearly coupled random oscillators. The approach rests on explicit bounds for the space derivatives of solutions, obtained via energy-like estimates. Bounds for the numerical approximations of solutions are given, and improved (sometimes appreciably) by means of an ‘a posteriori error analysis’. Plots are shown to illustrate the performance of the method, and comparison with a finite difference approach is also made.

Key Words: spectral method, nonlinear parabolic equations, integro-differential parabolic equations, populations of coupled oscillators, Kuramoto–Sakaguchi equation

Received 11 October 1999. Accepted 27 March 2000.


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