Highly oscillatory quadrature and low-regularity integrators for nonlinear evolution equations



7/06/2022 - 13:00 Georg Maierhofer (Sorbonne Université ) IMAG 106

Highly oscillatory integrals arise in a plethora of numerical methods for the simulation of physical systems, including in wave scattering and quantum physics. This has motivated a tremendous ef- fort over the recent decades to develop efficient algorithms for the approximation of such highly oscillatory integrals. Amongst them are so-called Filon methods, which combine ideas from classical quadrature with asymptotic theory to approximate these integrals reliably accurately and at uniform cost in frequency. However, one of the central difficulties encountered in Filon methods is the fast computation of the quadrature moments.
In this talk, we will describe a novel approach to the solution of this ‘moment-problem’ by providing a general framework for the construction of moment recursions that can achieve this desired task for a wide range of oscillators. In addition, based on a deep connection between oscillatory phenomena and the regularity of solutions to partial differential equations on periodic domains, we will see how Filon methods can be used in time-stepping methods to accurately capture frequency interactions in nonlinear evolution equations. In connection with recent advances in resonance-based methods, this insight allows us to design innovative numerical schemes, which can efficiently approximate low-regularity solutions to nonlinear systems, even when classical methods (such as exponential integrators) fail. We will see both theoretical results and numerical experiments which demonstrate the favourable performance of the proposed methods. This is joint work with Arieh Iserles, Nigel Peake and Katharina Schratz.