Speaker
Description
A large toolbox of numerical schemes to approximate the time dynamics in nonlinear PDEs has been meanwhile 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 nature of the solution, and this may lead to severe instabilities and loss of convergence. Nevertheless non-smooth phenomena play a fundamental role in modern physical modeling, e.g., singularity and shock formation, turbulences, etc., which makes it an essential task to develop suitable numerical schemes which capture effectively their dynamics.
In this talk I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying structure of resonances of nonlinear PDEs into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties even down to very low regularity.
