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Numerical solution of the small dispersion limit of Korteweg-de Vries and Whitham equations. (English) Zbl 1139.65069

The authors review the Lax-Levermore theory and the Whitham equations for the case of small dispersion in the Korteweg-de Vries (KdV) equation, along with the developments of S. Venakides [ibid. 43, No. 3, 335–361 (1990; Zbl 0705.35125)] and F. R. Tian [ibid. 46, No. 8, 1093–1129 (1993; Zbl 0810.35114)]. They then report on their method of numerical solution of the equations (through discrete Fourier transform etc., for the case of periodic boundary conditions) where the aim is to get solutions to machine accuracy. They are able to do so for the interval of [ 0.001,0.1] of the dispersion – which enables them to check the asymptotic formulae. The outcome is, e.g., that the oscillations of the KdV-solution start ealier than the gradient catastrophe and are present at a larger x-interval than the asymptotic theory (“Whitham zone”) predicts.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)

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