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• Klein, C. 2008. Fourth Order Time-Stepping for Low Dispersion Korteweg-de Vries and Nonlinear Schrödinger Equations.

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 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 Schrödinger 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.

• Klein, C.; Sparber, C.; & Markowich, P. 2007. Numerical Study of Oscillatory Regimes in the Kadomtsev–Petviashvili Equation.

The aim of this paper is the accurate numerical study of the Kadomtsev–Petviashvili (KP) equation. In particular, we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end, we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey–Stewartson system. In a second step, we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg–de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.

Lecturing & Teaching

I supervise graduate students at the Institut de Mathématiques de Bourgogne, UBFC. If you are interested in working with me, please get in touch.

For more details on my teaching and lecturing, please visit this CV page.