I am a Professor of Mathematical Physics at the Institut de Mathématiques de Bourgogne, UBFC, in Dijon, France.
My research interests include numerical methods and highly oscillatory PDEs. Here you’ll find resources and links to my publications.
My main contribution to the field of mathematical physics is in the application of modern, highly efficient numerical methods to the areas of differential equations, integrable systems, Riemann surfaces, general relativity, and various physical problems.
I am currently working on a project to improve medical imaging in Electrical Impedance Tomography (EIT).
Originally from Germany, I’ve been a professor at UBFC since 2007.
Here is my full CV.
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
For more details on my teaching and lecturing, please visit this CV page.
Here is a video gallery to illustrate some past work.
Finally, here you’ll find links to download free software for mathematical physicists.
Institut de Mathématiques de Bourgogne
9 avenue Alain Savary, BP 47870
21078 DIJON Cedex
+33 (0)3 80 39 58 58
[ christian.klein at ubfc.fr ]