A linear focusing mechanism for dispersive and non-dispersive wave problems

A linear focusing mechanism for a wave-packet propagation in a non-periodic domain is explained here using Fourier-Laplace spectral theory. The global analysis method is used here to obtain numerical properties at each node of the full domain. In this work, we show the spectacular growth of an error-packet at a particular node and wavenumber, a phenomenon termed as focusing which depends on: (a) CFL number, Nc and (b) numerical method used for discretization. In this study, we have shown the focusing phenomenon for the numerical solution of the 1D convection equation and linearized rotating shallow water equation.