MiS Preprint Repository
We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation
Christian KleinAbstract
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.
- Received:
- 15.11.06
- Published:
- 15.11.06
- MSC Codes:
- 65M70, 65L05, 65M20
- Keywords:
- exponential time-differencing, Korteweg-de Vries, nonlinear Schrödinger equation