Numerical methods for elastic wave propagation

There is a great need for numerical methods which treat time-dependent elastic wave propagation problems. Such problems appear in many applications, for example in geophysics or non-destructive testing. In particular, seismic wave propagation is one of the areas where intensive scientific computation has been developed and used since the beginning of the 70’s, with the apparition of finite differences time domain methods (FDTD). Although very old, these methods remain very popular and are widely used for the simulation of wave propagation phenomena or more generally for the numerical resolution of linear hyperbolic systems. They consist in obtaining discrete equations whose unknowns are generally field values at the points of a regular mesh with spatial step h and time step Δt. A prototype of these methods is the famous Yee§s scheme introduced in 1966 for Maxwell§s equations. There are several reasons that explain the success of Yee type schemes, among which their easy implementation and their efficiency which are related to the following properties: a uniform regular grid is used for the space discretization, so that there is a minimum of information to store and the data to be computed are structured: in other words, one avoids all the complications due to the use of non uniform meshes.an explicit time discretization is applied: no linear system has to be solved at each time step.