Multi-scale numerical methods
We are
developing simulation tools based on multi-scale numerical methods such as the
finite element method. The finite element method discretizes the governing
equations on an unstructured mesh with a variable resolution. It is thus
possible to locally change the resolution according to the scale of the
processes of interest. The flexibility of unstructured meshes allows us to
represent a broader range of spatial scales and thus make best use of the
computing resources. These developments are performed in the framework of the SLIM
project. SLIM is an unstructured-mesh hydrodynamical model developed at UCL and
that is currently used for a number of applications, including the study of the
connectivity in the Great Barrier Reef or the integrated modelling of the
land-sea continuum in complex river-estuary systems. Furthermore, we are also
interested in the mathematical properties of these models, like wave-dispersion
properties or stability of turbulence closure schemes.
Non-Brownian dispersion models
We are using stochastic models to
simulate the dispersion of tracers in complex systems, such as turbulent flows
or heterogeneous porous media. These models are not based on the Brownian-motion
assumption but instead assume that particles are following a Levy process.
Clouds of particles can therefore be represented by Levy distributions, which
form a family of so-called "heavy-tailed" probability distributions.
Such distributions are now used in a number of applications in finance,
hydrology or biology to represent the effect of large variations or
"extreme events" that can occur in complex systems. It can be shown
that Levy distributions are solutions of a fractional-order diffusion equation
that requires some specific numerical treatment. The non-local nature of
fractional-order diffusion operators has lead us to consider non-local
numerical methods, like the spectral element method, to solve that
equation.
