Space discretization using the wavelet-Galerkin Method for the solution of the acoustic wave equation in seismic modeling

The use of wavelet-based numerical schemes has become popular in the last three decades, especially for problems with local high gradients. Wavelets have several properties that are quite useful for representing solutions of partial differential equations (PDEs), such as orthogonality, compact support and exact representation of polynomials of a certain degree. The present work discusses an alternative to the usual finite difference (FDM) approach to the acoustic wave equation modeling by using a space discretization scheme based on the Galerkin Method. The combination of this method with wavelet analysis results in the Wavelet Galerkin Method (WGM) which has been adapted for the direct solution of the wave differential equation in a meshless formulation. This work also introduces Deslauriers-Dubuc wavelets (Interpolets) as interpolating functions. For a given order, Interpolets present the highest number of vanishing moments among all wavelet families. Examples in 1-D and 2-D were formulated using a central difference (second order) scheme for time differentiation. The main improvement in the presented formulation was the recognition of a different dispersion pattern when comparing FDM and WGM results using the same space and time grid.