Stability issues in discretization of wave equation

This paper discusses some of the interesting properties of stability analysis of a discretized wave equation. The solutions of the wave equation are wave functions, hence oscillating, so when testing stability the discretization scheme usually shows marginal stability. Marginal stability is a sufficient condition for a discrete scheme convergence and many authors don´t bother with mathematical consistency. However, inadequatly chosen discretization method may lead to the additional unwanted oscillations. This paper illustrates this effect in a different approach. First, the wave equation is introduced together with a Perfectly matched layer (PML). Then the 1D wave equation is discretized by using Finite Differences Method (FDM) and Finite-differences Time-domain method (FDTD). It is shown that the latter method does not produce spurious oscillations in the solution. Eigenvalue analysis is done to explain this effect and discuss stability of the numerical scheme.