From short to long time scales: Resonances and Structure in Computation

forthcoming

Symmetries, energy and mass conservation, integrability, measure invariance, scattering, etc. are major features in nonlinear systems and intrinsically shape their global dynamics. To guarantee a reliable numerical description of nonlinear phenomena on short and long time scales, faithful to the physical interpretation of differential equations, it is therefore crucial to preserve these structures at the discrete level. Otherwise important information becomes lost, hindering genuine long time simulations. For partial differential equations (PDEs) a major stumbling stone are numerical resonances, a purely discrete artefact, which destroy the geometric structure of the continuous problem and limit reliable computations to short time scales. This gets even worse for rough data: The rougher the solution, the less numerical resonance effects can be controlled and nothing is currently known how to overcome this. Non-smooth phenomena play however a fundamental role in modern physical modelling, e.g., singularity and shock formation, turbulence, etc., making it crucial to develop suitable numerical schemes which capture effectively their long-time behaviour. Positioned at the exciting interface of theoretical and computational PDE this project takes the challenging step from short to long time scales in the computation of nonlinear PDEs by tackling their time dynamics in a nonlinear manner. This will allow us to tame the numerical resonances and achieve reliable computations up to very long time scales, and down to very low regularity. The ambitious approach will shape the computation of PDEs, as linearisation is not sufficient to reproduce global dynamics, especially once roughness comes into force, and offers an immense range of prospective applications. Roughness and long time scales are everywhere in nature: from singularity formation in general relativity to shock formation in fluid mechanics, from rogue waves in oceanography to propagation of signals in fibre optics.