## Section: New Results

### Higher Order Absorbing Boundary Conditions for the Wave Equation in Discontinuous Galerkin Schemes

Participants : Hélène Barucq, Julien Diaz, Véronique Duprat.

The numerical simulation of waves propagation generally involves boundary conditions which both represent the behavior of waves at infinity and provide a mathematical tool to define a bounded computational domain in which a finite element method can be applied. Most of these conditions are derived from the approximation of the Dirichlet-to-Neumann operator and when they both preserve the sparsity of the finite element matrix and enforce dissipation into the system, they are called absorbing boundary conditions. Most of the approximation procedures are justified into the hyperbolic region which implies that only the propagative waves are absorbed. If the exterior boundary is localized far enough from the source field, the approximation is accurate and the absorbing boundary condition is efficient. However, the objective is to use a computational domain whose size is optimized since the solution of waves problems requires to invert matrices whose order is very large and is proportional to the distance between the source field and the exterior boundary. Hence it is a big deal to derive absorbing boundary conditions which are efficient when the exterior boundary is close to the source field and it is necessary to construct conditions which are efficient not only for propagative waves but both for evanescent and glancing waves. In a recent work [43] , new conditions have been derived from the modal analysis of the wave equation set in the neighborrhood of a prolate spheroidal boundary. From the numerical analysis of the error, it has been proven [44] that these conditions are efficient for each type of waves and then, they outperform the current absorbing conditions. However, the derivation procedure in [43] is based on the representation of the Helmholtz equation in an elliptic coordinate system reproducing the geometry of the exterior boundary and it is not obvious how to generalize the conditions to an arbitrarily-shaped boundary. Recently, a new condition has been derived [65] from an approximation of the Dirichlet-to-Neumann operator which is valid both for propagative and evanescent waves and it extends the condition which was formerly proposed by Higdon [67] . By using a classical finite element scheme, Hagstrom et al. [65] have shown the improvements induced by the new condition. We have then addressed the issue of integrating the new condition into a DG scheme which is more appropriate to reproduce the propagation of waves into heterogeneous media and we have observed numerical instabilities [36] . We have next considered optimized ABCs adapted to arbitrarily-shaped regular boundaries and we have constructed a transparent condition based on the decomposition of the exact solution into a propagating field, an evanescent field and a grazing field. Then, a new condition is obtained by combining the approximations of the transparent condition in the three corresponding regions (hyperbolic, elliptic and glancing regions). It is not classical in the sense that it involves a fractional derivative arising from the grazing part of the solution. However, the condition is easily included into a finite element scheme and we have implemented it into an Interior Penalty Discontinuous Galerkin formulation. Numerical experiments have been performed and the results have shown that it does not modify the CFL condition. Furthermore, the absorption rate is improved when compared to classical ABCs.

Recently, we have enriched the ABC by including a condition for grazing waves. The condition involves a fractional derivative of the curvilinear abscissa in the sense of Caputo. It is discretized by a quadrature formula where the weights can be neglected outside a small neighborhood of each point of the surface. The discretized condition is thus pseudo-local. A new ABC has then be constructed by applying it to the initial one. Since it is pseudo-local, it does not break down the local property of the initial condition and it does not modify the long-time behavior.

Our results have been presented to the peer-reviewed WAVES 2009 (Pau, June 2009) and ICTCA (Dresden, September 2009) conferences. A paper has been submitted