Project description
New approach to accurately simulating long-time behaviour of non-linear systems
Non-linear systems are governed by symmetries, energy conservation and integrability, which shape their global dynamics. To reliably simulate these systems over time, it is crucial to preserve these structures at the discrete level. A major stumbling block for partial differential equations (PDEs) are numerical resonances that limit reliable computations to short timescales. The rougher the solution, the harder to control numerical resonance effects. The ERC-funded ReStruCt project will help overcome this by developing suitable numerical schemes that effectively capture the long-time behaviour of non-linear PDEs. The ambitious approach will shape PDE computation, as linearisation is not sufficient to reproduce global dynamics, especially once roughness comes into force. By enabling accurate long-term simulations, ReStruCt could advance understanding of complex phenomena in general relativity and signal propagation.
Objective
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.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
- natural sciences earth and related environmental sciences oceanography
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Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2025-COG
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75006 Paris
France
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