Nonlinear Fourier integrators for dispersive equations
A talk in the Oberseminar Numerik series by
Katharina Schratz from Heriot Watt University
| Abstract: | A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization
techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the
full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical
schemes allow a precise and efficient approximation. This, however, drastically changes whenever "non-smooth"
phenomena enter the scene such as for problems at low-regularity and high oscillations. Classical schemes fail to
capture the oscillatory parts within the solution which leads to severe instabilities and loss of convergence.
In this talk I present a new class of nonlinear Fourier integrators. The key idea in the construction of the new
schemes is to tackle and deeply embed the underlying structure of resonances into the numerical discretization.
As in the continuous case, these terms are central to structure preservation and offer the new schemes strong
geometric structure at low regularity. |