Section: New Results
Higher Order Absorbing Boundary Conditions for the Wave Equation applied to Discontinuous Galerkin Methods
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  , 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  that these conditions are efficient for each type of waves and then, they outperform the current absorbing conditions. However, the derivation procedure in  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  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  . By using a classical finite element scheme, Hagstrom et al.  have shown the improvements induced by the new condition. In this work, we intend to analyze if the new condition can be introduced into a discontinuous Galerkin method which is more accurate to reproduce the propagation of waves into heterogeneous media . To analyze the impact of the new condition on the accuracy of the numerical solution, we also consider the Higdon condition and we are able to compare the efficiency of the two conditions when used into a discontinuous Galerkin approximation.