Solution of Wave Equation in Rods Using the Wavelet-Galerkin Method for Space Discretization

Abstract

The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs) in the last three decades. Therefore, the use of wavelets scale functions as a basis in computational analysis holds some promise due to their compact support, orthogonality, localization and multiresolution properties. 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 using scale functions 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 paper presents an extension of previous works which dealt with linear elasticity problems. This work also introduces Deslauriers-Dubuc scaling functions (also known as Interpolets) as interpolating functions in a Galerkin approach considering wave propagation problems. Examples in 1-D were formulated using a central difference (second order) scheme for time differentiation. Encouraging results were obtained when compared with the FDM using the same time steps. 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.