## Section: New Results

### Numerical methods combining local time stepping and mixed hybrid elements for the terrestrial migration

Participants : Caroline Baldassari, Hélène Barucq, Henri Calandra [ Expert Engineer, TOTAL ] , Bertrand Denel [ Research Engineer, TOTAL ] , Julien Diaz.

We work on the development of a software for the Reverse Time Migration of acoustic waves which combines mixed hybrid elements in space and a local time stepping scheme. The discretization in space allows us to take the topography into account, which usually outperforms finite difference schemes. In order to illustrate this point, we have compared the solution obtained with a Dicontinuous Galerkin Finite Element Method (DGFEM) to the one obtained with finite difference methods or with the GSP method and we have shown that DGFEM gives much more accurate results. We have also compared the solution obtained with DGFEM to the one obtained with spectral element methods, which requires the use of quadrilateral (in 2D) of hexaedral (in 3D) cells and is therefore less convenient to deal with a topography. Both solution present the same order of accuracy, which encourage us to use DGFEM to deal with a topography.

Near the topography (or in very thin layers), the cells of the mesh have to be very small compared to the cells far from the topography. In such configuration, it is not useful to user high order elements in the whole domain and we propose to use second order elements in the fine part of the mesh and high order elements in the coarse part. The numerical experiments we have performed show actually that there is no difference between the solutions obtained with meshes composed only of high-order cells and with meshes composed of second order cells near the topography. Moreover the use of second order cells reduces dramatically the computational burden.

Since we use explicit time-schemes such as Leap-Frog scheme (for second order time-discretization) or Modified Equation scheme (for higher-order time discretization), the time step is contrained by the so-called CFL (Courant-Friedrichs-Lewy) condition, which is dictated by the smallest cell of the mesh. If the mesh contains both very small and very large cells, it is necessary to use a local-time stepping strategy as the one proposed by [27] . Moreover, if the mesh contains cells of various order, it is useful to adapt the order of the time-discretization to the space discrization. We have then extended the local time-stepping method to handle different order of time-discretization, and we have tested the new method in 1D and in 2D. The 1D tests showed that this method does not hamper the accuracy of the space discretization while it allows to decrease the computational burden. The 2D experiments showed that this method could be implemented in the Reverse Time Migration algorithm. The results of the year have been presented in 4 peer-reveiewed conferences which are listed below.