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Low-regularity and high oscillations: numerical analysis and computation of dispersive evolution equations

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.

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.

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Topic(s)

Call for proposal

ERC-2019-STG

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Funding Scheme

ERC-STG - Starting Grant

Coordinator

SORBONNE UNIVERSITE
Net EU contribution
€ 1 499 905,00
Address
21 rue de l'ecole de medecine
75006 Paris
France

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Region
Ile-de-France Ile-de-France Paris
Activity type
Higher or Secondary Education Establishments
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Other funding
€ 0,00

Beneficiaries (2)

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Last update: 15 September 2022
Record number: 228586

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European Union, 2023