Numerical Analysis of the Two Dimensional Wave Equation : Using Weighted Finite Differences for Homogeneous and Hetrogeneous Media

Abstract: This thesis discusses properties arising when finite differences are implemented forsolving the two dimensional wave equation on media with various properties. Both homogeneous and heterogeneous surfaces are considered. The time derivative of the wave equation is discretised using a weighted central difference scheme, dependenton a variable parameter gamma. Stability and convergence properties are studied forsome different values of gamma. The report furthermore features an introduction to solving large sparse linear systems of equations, using so-called multigrid methods.The linear systems emerge from the finite difference discretisation scheme. Aconclusion is drawn stating that values of gamma in the unconditionally stable region provides the best computational efficiency. This holds true as the multigrid based numerical solver exhibits optimal or near optimal scaling properties.