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## Section: New Results

### Discontinuous Galerkin methods for the elastodynamic equations

#### DGTD-${ℙ}_{p}$ method for the elastodynamic equations

Participants : Nathalie Glinsky, Fabien Peyrusse.

We continue developing high order non-dissipative discontinuous Galerkin methods on simplicial meshes for the numerical solution of the first order hyperbolic linear system of elastodynamic equations. These methods share some ingredients of the DGTD-${ℙ}_{p}$ methods developed by the team for the time domain Maxwell equations among which, the use of nodal polynomial (Lagrange type) basis functions, a second order leap-frog time integration scheme and a centered scheme for the evaluation of the numerical flux at the interface between neighboring elements. Recent results concern two particular points.

The first novelty is the extension of the DGTD-${ℙ}_{p}$ method initially introduced in [5] to the numerical treatment of viscoelastic attenuation. For this, the velocity-stress first order system is completed by additional equations for the anelastic functions describing the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. In practice, one needs to add 3L additional equations in 2D and 6L in 3D, where L is the number of relaxation mechanisms of the generalized Maxwell body. This method has been implemented in 2D and validated thanks to comparisons with a FDTD method.

The second contribution is concerned with the numerical assessment of site effects especially topographic effects. The study of measurements and experimental records proved that seismic waves can be amplified at some particular locations of a topography. Numerical simulations are exploited here to understand further and explain this phenomenon. The DGTD-${ℙ}_{p}$ method has been applied to a realistic topography of Rognes area (where the Provence earthquake occured in 1909) to model the observed amplification and the associated frequency. Moreover, the results obtained on several homogeneous and heterogeneous configurations prove the influence of the medium in-depth geometry on the amplifications measures at the surface [26] , [25] .