Project description DEENESFRITPL Low-regularity and high oscillations: computation of dispersive equations (LAHACODE) Partial differential equations (PDEs) play a central role in mathematics, allowing us to describe physical phenomena ranging from ultra-cold atoms up to ultra-hot matter, from learning algorithms to fluids in the human brain. To understand nature we have to understand their qualitative behavior and compute reliably their numerical approximation. While linear problems and smooth solutions are nowadays well understood, a reliable description of ‘non-smooth’ phenomena remains a challenging open problem. The overall ambition of the ERC-funded project LAHACODE is to make a crucial step towards closing this gap by deeply embedding the underlying structure of resonances into the numerical discretisation. This will allow us to link the finite dimensional discretisation to powerful existence results for nonlinear PDEs at low regularity. Show the project objective Hide the project objective Objective Partial differential equations (PDEs) play a central role in mathematics, allowing us to describe physical phenomena ranging from ultra-cold atoms (Bose–Einstein condensation) up to ultra-hot matter (nuclear fusion), from learning algorithms to fluids in the human brain. To understand nature we have to understand their qualitative behavior: existence and long time behavior of solutions, their geometric and dynamical properties – as well as to compute reliably their numerical solution. While linear problems and smooth solutions are nowadays well understood, a reliable description of ‘non-smooth’ phenomena remains one of the most challenging open problems in computational mathematics since the underlying PDEs have very complicated solutions exhibiting high oscillations and loss of regularity. This leads to huge errors, massive computational costs and ultimately provokes the failure of classical schemes. Nevertheless, ‘non-smooth phenomena’ play a fundamental role in modern physical modeling (e.g. blow-up phenomena, turbulences, high frequencies, low dispersion limits, etc.) which makes it an essential task to develop suitable numerical schemes. The overall ambition of LAHACODE is to make a crucial step towards closing this gap – addressing the fundamental question: How and to what extent can we reproduce the qualitative behavior of differential equations in a finite (discretized) world? LAHACODE is situated at the challenging frontiers of analysis and numerics. The main objective is to develop a novel class of numerical schemes for nonlinear PDEs with strong geometric structure at low regularity and high oscillations. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying structure of resonances in the numerical discretizations. As in the continuous case, these terms are central to structure preservation, and provide the new schemes with remarkable properties – allowing reliable approximations where classical schemes fail. Fields of science natural sciencesphysical sciencesnuclear physicsnuclear fusionnatural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equationsnatural sciencesmathematicsapplied mathematicsnumerical analysis Keywords MSC2010: 65Mxx Partial differential equations initial value and time-dependent initial-boundary value problems Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2019-STG - ERC Starting Grant Call for proposal ERC-2019-STG See other projects for this call Funding Scheme ERC-STG - Starting Grant Host institution SORBONNE UNIVERSITE Net EU contribution € 1 499 905,00 Address 21 RUE DE L'ECOLE DE MEDECINE 75006 Paris France See on map Region Ile-de-France Ile-de-France Paris Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 499 905,00 Beneficiaries (2) Sort alphabetically Sort by Net EU contribution Expand all Collapse all SORBONNE UNIVERSITE France Net EU contribution € 1 499 905,00 Address 21 RUE DE L'ECOLE DE MEDECINE 75006 Paris See on map Region Ile-de-France Ile-de-France Paris Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 499 905,00 LUDWIG-MAXIMILIANS-UNIVERSITAET MUENCHEN Participation ended Germany Net EU contribution € 0,00 Address GESCHWISTER SCHOLL PLATZ 1 80539 Muenchen See on map Region Bayern Oberbayern München, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost No data
