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Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations. (English) Zbl 1186.65134
Summary: Purely dispersive equations, such as the Korteweg-de Vries and the nonlinear Schrödinger 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 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 equations in the small dispersion limit: an exponential time-differencing fourth-order Runge-Kutta method as proposed by S. M. Cox and P. C. Matthews [J. Comput. Phys. 176, No. 2, 430–455 (2002; Zbl 1005.65069)] in the implementation by A.-K. Kassam and L. N. Trefethen [SIAM J. Sci. Comput. 26, No. 4, 1214–1233 (2005; Zbl 1077.65105)], integrating factors, time-splitting, B. Fornberg and T. A. Driscoll’s ‘sliders’ [J. Comput. Phys. 155, No. 2, 456–467 (1999; Zbl 0937.65109)], and an ordinary differential equation solver in Matlab.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65L05 Numerical methods for initial value problems involving ordinary differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
Software:
Matlab
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