High Order Symmetric Finite Difference Schemes for the Acoustic Wave Equation

Abstract: When someone speaks, or when a piano is played, sound waves are spread through the room. Physically the propagation of sound is governed by the acoustic wave equation. This thesis deals with the problem of finding solutions to this equation by numerical methods. These methods utilize the tremendous power of the modern computer to perform arithmetic operations.The two most common methods for the solution of a partial differential equation like the wave equation are the finite element (FEM) and the finite difference method (FDM). Traditionally the FEM is the method of choice for stationary problems in structural mechanics, while the FDM is often preferred for time dependent problems. In this study we combine the spatial discretization of a specific FEM on a uniform grid with a time discretization from a FDM. The new scheme is explicit, symmetric and has fourth order of accuracy in space and time for Neumann and/or Dirichlet boundary conditions on a rectangle.The Helmholtz equation is employed for the analysis of the spatial discretization. Concerning the time dependence, we focus on problems for which an explicit time marching scheme is the most adequate choice from a physical point of view.When the FEM is applied to a time-dependent problem, a mass matrix appears in the front of the time evolution operator. To avoid the costly solution of a sparse linear system for each time step, the mass matrix is approximated by a diagonal matrix (mass lumping). For a particular basis of piecewise cubic polynomials we show that mass lumping with the rowsum preserves the order of accuracy for problems with Neumann boundary conditions. For Dirichlet boundary conditions mass lumping must be done differently in the vicinity of the boundaries to attain full accuracy. The theory of FDM is used for analysis of accuracy and stability.We study the performance of the new scheme in the presence of media discontinuities. It is found that the continuity of pressure and particle velocity is fulfilled to the first order, even though the discontinuities are not aligned with the cartesian grid.Numerical examples are presented which illustrate the outstanding performance of the new scheme.