Geodesic
Fourth order time-stepping for low dispersion Korteweg - de Vries and nonlinear Schrödinger equations
Electronic transactions on numerical analysis, Tome 29 (2008).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Purely dispersive equations, such as the Korteweg-de Vries and the nonlinear Schr$\ddot $odinger equations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blowup. Fourth order time-stepping in combination with spectral methods is beneficial to numerically resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Korteweg-de Vries and the focusing and defocusing nonlinear Schr$\ddot $odinger equations in the small dispersion limit: an exponential time-differencing fourth-order Runge-Kutta method as proposed by Cox and Matthews in the implementation by Kassam and Trefethen, integrating factors, time-splitting, Fornberg and Driscoll's `sliders', and an ODE solver in Matlab.
Classification : 65M70, 65L05, 65M20
Keywords: exponential time-differencing, korteweg-de Vries equation, nonlinear schr$\ddot $odinger equation, split step, integrating factor 
@article{ETNA_2008__29__a6,
     author = {Klein, Christian},
     title = {Fourth order time-stepping for low dispersion {Korteweg} - de {Vries} and nonlinear {Schr\"odinger} equations},
     journal = {Electronic transactions on numerical analysis},
     publisher = {mathdoc},
     volume = {29},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2008__29__a6/}
}
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JO  - Electronic transactions on numerical analysis
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VL  - 29
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UR  - http://geodesic.mathdoc.fr/item/ETNA_2008__29__a6/
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%T Fourth order time-stepping for low dispersion Korteweg - de Vries and nonlinear Schrödinger equations
%J Electronic transactions on numerical analysis
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Klein, Christian. Fourth order time-stepping for low dispersion Korteweg - de Vries and nonlinear Schrödinger equations. Electronic transactions on numerical analysis, Tome 29 (2008). http://geodesic.mathdoc.fr/item/ETNA_2008__29__a6/