Section: New Results
A new modified equation approach for solving the wave equation
In this work, we construct new fourth order schemes in space and time for the wave equation by applying the modified equation technique in an original way. Indeed, most of the works devoted to the solution of the wave equation consider first the space discretization of the system before addressing the question of the time discretization. The idea is here to invert the discretization process by applying first the time discretization thanks to the modified equation technique and after to consider the space discretization. After the time discretization, an additional bilaplacian operator, which can not be discretized by the classical finite elements appears. If the acoustic medium is homogeneous or has smooth heterogeneities, the solution is C1 and therefore we have to consider C1 finite elements (such as the Hermite ones) or Discontinuous Galerkin finite elements (DGFE) whose C1 continuity is enforced through an appropriate penalty term. In a strongly heterogeneous media, the solution is no longer C1 because of the discontinuities of the physical parameters and Hermite elements are not adapted to this problem. DGFE can however be used by imposing the continuity of a suitable physical quantity (corresponding to the classical transmission conditions) instead of the C1 continuity. Moreover, this technique can be extended to obtained arbitrarily even high-order schemes by considering p-laplacian operators and we have also considered the sixth order scheme. We have compared the solution obtained by the new method with DGFE to the one obtained with a classical second order method (using also DGFE) in 1D and 2D first in a simple homogeneous medium then in a heterogeneous bilayered medium. Numerical results showed that the scheme with the bilaplacian (resp. 3-laplacian) operator is actually fourth (resp. sixth) order accurate. Moreover it appeared that the stability condition is not penalized, in the sense that the stability conditions of the 4th and the 6th order schemes are very close to the one of the classical Leap-Frog scheme (actually it is even a little higher). This means that the new schemes provide high accuracy for the same cost than the classical second-order scheme.
Now, we are considering the adaptation of the order of the scheme in the various cells of the mesh. To do so, we consider for instance a scheme with bilaplacian on the coarse cells of a given mesh and a classical leap-frog scheme on the fine cells. To perform the space discretization, we have to take into account appropriate transmission conditions at the interface between the fine and the coarse mesh. Premilinary results in 1D shows that the accuracy of the global scheme is not hampered provided that the fine cells are small enough compared to the coarse mesh.
Our results have been presented to the peer-reviewed WAVES 2009 (Pau, June 2009) and ICTCA (Dresden, September 2009) conferences.