The team Magique-3D works on the numerical simulation of wave propagation to represent geophysical phenomena. It mainly develops optimized algorithms for seismic imaging problems and seismology but it works also on scattering problems for which it addresses modelling issues such as the construction of efficient boundary conditions. This question is less critical for seismic imaging problems than for scattering problems and seismology because in the first case, the goal is to produce an image which is obtained from an imaging condition that is not sensitive to the boundary effects. On the contrary, the solution of scattering problems can be drastically polluted by the spurious effects of the boundary.
The post-doctoral work that Magique-3D is proposing will be to develop in the framework of a research program that we are considering for a long time. Two finite element codes are thus available for acoustic and elastic waves and the work will be done in collaboration with the leaders of the codes. The numerical simulation of wave propagation generally involves the coupling of wave equations with boundary conditions that are set on the external boundary of the computational domain. There exist plenty of boundary conditions which are generally called Absorbing Boundary Conditions (ABC) and they obviously differ from the method that is used for their construction. In case of time-dependent problems and for arbitrarily-shaped boundaries, the existing conditions are representing propagating waves only. They are thus not very efficient in many situations, for instance in case of multiple scatterers where grazing rays are very significant. Hence there is a need of constructing new ABCs that are able to take the full wave into account. In the team Magique-3D, we have developed new ABCs for the acoustic wave equation. They involve a fractional derivative and they are easy to carry out inside a numerical scheme combining a discontinuous Galerkin approximation with high-order time schemes. The numerical experiments that have been performed show that the new condition improves the accuracy of the numerical field. Nevertheless, the mathematical study of the boundary value problem remains to be done and the long-time stability of the problem has been observed numerically only. The post-doctoral researcher work will consist in addressing the above issues and to test the new condition in 3D. If there is time enough, the case of electromagnetic waves will be considered, at least from a theoretical point of view.
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