Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation

Christian Klein

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Submission date: 15. Nov. 2006
Pages: 18
published in: Electronic transactions on numerical analysis, 29 (2008), p. 116-135 
Bibtex
MSC-Numbers: 65M70, 65L05, 65M20
Keywords and phrases: exponential time-differencing, Korteweg-de Vries, nonlinear Schrödinger equation
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Abstract:
Purely dispersive equations as the Korteweg-de Vries and the nonlinear Schrödinger equation in the limit of small dispersion have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. Fourth order time-stepping in combination with spectral methods is beneficial to numerically resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Korteweg-de Vries and the focusing and defocusing nonlinear Schrödinger in the small dispersion limit: an exponential time-differencing fourth-order Runge-Kutta method as proposed by Cox and Matthews in the implementation by Kassam and Trefethen, integrating factors, time-splitting, Fornberg and Driscoll’s ‘sliders’, and an ODE solver in Matlab.