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PhD Position: Fast and robust solvers for non-symmetric systems arising in the discretizations of underground flows

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Required Education : MSc or equivalent
Start date : 1 October 2019
Mission duration : 36 months

Scientific context and objectives

The goal of the project is the development of fast and robust solvers for matrices arising in the discretization of partial differential equations governing subsurface flows in porous media. The material parameters in the equation are discontinuous and may be constant in time (the linear case) or may depend on the flow (the non-linear case). An (elliptic) equation related to the hydraulic head has to be solved and as many transport equations as tracers to study. The main issue to tackle is the fast and robust resolution of the transport equations. The considered application will require more than 200*106 elements to describe a complex geometry for the spatial approximation. The spatial discretization is done with a CDO vertex-based discretization (lowest order discretization). For the linear case of the material parameters, a backward Euler time step or potentially Crank-Nicolson are used for the temporal discretization. In each time step, a linear system with as many unknowns as there are vertices in the mesh must be solved. If the material parameters depend on the current flow, then the problem becomes non-linear. This case is even more difficult, as on top of the time stepping scheme, there are additional iterations for a non-linear solver. In one simulation, we can expect to solve several thousands of large linear systems.

The calculations at EDF are performed with Code_Saturne, which has already demonstrated excellent scaling behavior. At the time of writing, the scientific literature does not contain any result that demonstrates robustness and scalability of any linear solver for the CDO discretisation of transport equations. This question has therefore scientific and industrial relevance as the linear solvers dominate wall clock time in time-dependent simulations on large meshes. In the near future, CDO face-based schemes (closely related to the lowest order HHO discretization) could also be used to discretize the transport equations. In this case, the degrees of freedom are located at mesh faces and there is no robust and scalable solver available even for the diffusion part.

This topic combines aspects in HPC, parallel numerical algorithms and multigrid methods.


Required qualifications and skills:

We are looking for a candidate with a Master (or equivalent) in the fields of computational fluid dynamics or applied mathematics and who should have interest in numerical algorithms and iterative solvers.


Place of work and industrial partnership:

CERFACS is a basic and applied research center, specialized in modeling and numerical simulation. Through its facilities and expertise in High Performance Computing, CERFACS deals with major scientific and technical research problems of public and industrial interest. The PhD student will be integrated into the parallel algorithm team led by Professor Ulrich Ruede. The research will be done in an industrial partnership with EDF R&D.


To apply, please send your CV and a cover letter to

Carola Krusecarola.kruse@cerfacs.fr
Frank Huelsemannfrank.hulsemann@edf.fr