This PhD project deals with sparse
grid methods for time-dependent PDE problems.
Advection-diffusion-reaction problems from
air quality modeling form an important test case.
Joint project with MAS2.
The project started in January 1998. The planned duration is four years.
Project Description:
The aim of this 4-year PhD project
is to make significant progress in the numerical solution of
large-scale transport problems: systems of partial differential equations
of the advection-diffusion-reaction type. Such problems are frequently
encountered in applications. They play a prominent role
in the mathematical modeling of pollution of
atmospheric air, surface water and groundwater.
Advanced models are three-dimensional
in space. Their 3D nature and the necessity of modeling
transport and chemical exchange between different components
over long time spans,
In the past, much research has been done on developing efficient
solvers, notably advection schemes, tailored integrators for stiff systems
of ordinary differential equations and other time stepping techniques.
This has already led to significant progress.
However, for advanced 3D modeling,
computer capacity (computing time and memory) still is a severe
limiting factor. This limitation is felt in particular in the area of
global air pollution modeling where the 3D nature always
leads to huge numbers of grid points in each of which many calculations must
be carried out.
This project aims at taking up a new line of research: application
of sparse-grid techniques in order to reduce the number of grid points,
of course without loss of accuracy and in combination
with an appropriate, efficient time stepping process.
Sparse grids were introduced by Zenger in the early nineties
to reduce the degrees of freedom in finite element calculations.
Recent literature
in the field of reactive flow problems and multigrid methods for stationary
differential equations confines this
very promising development.
Sparse grids can be considered through multivariate extrapolation
techniques. We plan to investigate these techniques in order
to further enhance the efficiency
of solving advection-diffusion-reaction problems.
By error analysis and experimentation,
the supposed advantage of sparse grids must be shown.
To assess the newly developed algorithms for truly practical
purposes, at the end use will be made of a prototype of a global atmospheric
air pollution model. Implementation aspects, including
vectorization and parallelization, must be taken into account
in this assessment.
People involved:
Last update 98/02/17.