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MiS Preprint

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

Christian Klein


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.

MSC Codes:
65M70, 65L05, 65M20
exponential time-differencing, Korteweg-de Vries, nonlinear Schrödinger equation

Related publications

2008 Journal Open Access
Christian Klein

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

In: Electronic transactions on numerical analysis, 29 (2008), pp. 116-135