Project description
How integrators could enhance computation of dispersive equations
A geometric integrator is a numerical method that preserves the geometric properties of the exact flow of a differential equation. Some of nature’s most fascinating phenomena, such as shock waves and the breaking of ocean waves at the shore, are mathematically best described using discontinuities (low regularity). However, there are few methods that can perform well in low-regularity regimes and at the same time preserve the geometric structure of the underlying differential equation. Funded by the Marie Skłodowska-Curie Actions programme, the GLIMPSE project will address this need for structure-preserving, low-regularity integrators for dispersive partial differential equations. If successful, numerical methods developed in GLIMPSE could be used to improve simulations used in weather forecasting and disaster prevention from extreme ocean events.
Objective
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.
Fields of science
Keywords
Programme(s)
- HORIZON.1.2 - Marie Skłodowska-Curie Actions (MSCA) Main Programme
Funding Scheme
HORIZON-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European FellowshipsCoordinator
75006 Paris
France