HEИндивидуална стипендия2022–2023

GLIMPSE · Geometric and Low-regularity Integrators for the Matching and Preservation of Structure in the computation of dispersive Equations

„Хоризонт Европа“ — Действия „Мария Склодовска-Кюри“

Период: 2022-05-01 → 2023-10-31
Финансиране от ЕС: 156 088 €
Участници: 1
Схема: HORIZON-TMA-MSCA-PF-EF

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Накратко на български

Математическите методи за симулиране на вълни, като тези при разбиването на океанските вълни, се анализират за по-голяма точност. Това помага за по-доброто изчисляване на сложни физични процеси, при които се появяват резки промени или ударни вълни.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Geometric and Low-regularity Integrators for the Matching and Preservation of Structure in the computation of dispersive Equations

Some of the most intriguing phenomena in nature arise when the underlying physical laws can be described using nonlinear dispersive partial differential equations. This means that waves of different frequencies travel at different speeds – a mechanism that is, for instance, responsible for the breaking of ocean waves near the shore. When a computer is asked to approximate solutions that exhibit discontinuities (low-regularity), as is the case for instance in shock waves, these nonlinear frequency interactions pose a significant challenge which has recently been addressed by the development of so-called resonance-based numerical schemes. Despite their success, ‘resonance-based’ schemes are a relatively recent development and many interesting open problems are yet to be studied, including their (in-)ability to preserve geometric structure from the underlying model. In this MSCA project the research fellow has, together with collaborators, successfully addressed this open problem through the construction of structure preserving low-regularity integrators. In particular, we have achieved the following major milestones: (i) The construction of symmetric low-regularity integrators for specific applications (the nonlinear Schrödinger equation, the Korteweg–de Vries equation and the isotropic Landau–Lifschitz equation); (ii) The classification of symmetric resonance-based schemes (low-regularity integrators) for a large class of dispersive nonlinear equations; (iii) The construction of symplectic low-regularity integrators for the Korteweg–de Vries equation and the one-dimensional nonlinear Schrödinger equation; (iv) The study of the long-time behaviour of symmetric resonance-based schemes for the nonlinear Schrödinger equation with weak nonlinearity and the design of a tailored integrator with significantly reduced error in the long-time regime; (v) An application of these new schemes to the simulation of the evolution of vortex filaments in ideal fluids.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

If mathematics is the language of physical sciences, differential equations are their grammar. Yet, to understand them, we need computational algorithms. Some of the most intriguing phenomena in nature arise when the underlying physical laws can be described using nonlinear dispersive partial differential equations. This means that waves of different frequencies travel at different speeds -- a mechanism that is, for instance, responsible for the breaking of ocean waves near the shore. When a computer is asked to approximate solutions that exhibit discontinuities (low-regularity), as is the case for instance in shock waves, these nonlinear frequency interactions pose a significant challenge which has recently been addressed by the development of so-called resonance-based numerical schemes. In many applications, it is desirable to apply geometric numerical integrators -- algorithms that preserve geometric structure of the underlying equation such as conservation of energy or time reversibility. However, there is only a very limited set of methods available that can address both challenges in unison, i.e. perform well in low-regularity regimes and preserve geometric structure of the underlying differential equation. Such algorithms, if more widely developed, would provide a valuable tool for a range of applications, including extreme events in ocean waves and atmospheric models. The goal of this proposed research is to address this need for structure-preserving low-regularity integrators for dispersive partial differential equations. The proposed project lies at the interface of computational mathematics, analysis and physical applications and, if successful, the results of this proposal have the potential to enhance both our current understanding of numerics for dispersive equations and, in the medium term, improve practical simulations which are used in weather forecasting and efficient disaster prevention from extreme ocean events.

Оригинален текст от CORDIS (на английски).

Участници

SORBONNE UNIVERSITE · ParisКоординаторФранция