DS10 - Défi des autres savoirs

Numerical boundaries and coupling – Nabuco

Submission summary

This project deals with the interplay between boundary conditions and transport or dispersive phenomena. It is motivated by the modeling of physical processes in which wave propagation is mainly governed by dispersion, be it encoded either in the underlying governing equations or in the chosen numerical discretization, and by wave breaking that comes from the nonlinear features of the physical process. The model equations that are being considered range from `toy' linear models such as the transport, Schrödinger or Airy equations, to complex, nonlinear systems such as the Euler-Korteweg, Boussinesq, Green-Naghdi or full water wave equations. The main target application of the project is coastal oceanography, that is, water wave reduced models in the weakly -or even fully- nonlinear shallow water regime, but the work plan will be extended to more general dispersive equations whenever possible.

Boundary effects may arise in a variety of contexts. There may be `fixed' physical boundaries (walls or pipe extremities in fluid flows, bottom of the ocean in gravitational surface wave modeling), `artificial' boundaries introduced for the purpose of numerical simulation, or even `moving', namely `free', boundaries arising in the modeling of compressible inviscid fluid flows (e.g., shock waves) or incompressible fluids submitted to gravity. Fixed and free boundaries also arise when coupling two models.

The NABUCO project is primarily of a numerical nature. It aims at studying the influence of boundary conditions on numerical schemes with the aim of designing accurate, computationally efficient, transparent/absorbing/transmission numerical conditions, specifically in coastal oceanography simulations. In this respect, the main target is to improve the reliability of numerical models to help forecasting extreme events such as tsunamis or hurricanes. With this ultimate computational goal in mind, the work plan incorporates a thorough analytic investigation of stability issues for numerical boundary conditions in discretized transport or dispersive equations, in order to give a sound basis to the design of accurate, yet computationally cheap and stable, boundary conditions in numerical codes.

The scientific objectives of the NABUCO project are therefore gathered around three main axes: (i) the analysis and implementation of discrete transparent/artificial boundary conditions for multidimensional dispersive systems, possibly incorporating small parameters, (ii) a unified analysis of boundary conditions for either continuous or discretized dispersive systems in a fixed domain, (iii) the coupling of hyperbolic and dispersive models in coastal oceanography. The project gathers participants with a strong expertise in physical modeling, analysis of partial differential equations and numerical schemes, up to scientific computing.

Project coordination

Jean-François Coulombel (Institut de mathématiques de Toulouse)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

ICJ Institut Camille Jordan
IMT Institut de mathématiques de Toulouse

Help of the ANR 257,601 euros
Beginning and duration of the scientific project: December 2017 - 48 Months

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