Fourier beyond dispersion: Wavenumber explicit and precise accuracy of FDMs for the Helmholtz equation

We propose a practical tool for evaluating and comparing the accuracy of FDMs for the Helmholtz equation. The tool based on Fourier analysis makes it easy to find wavenumber explicit order of convergence, and can be used for rigorous proof. It fills in the gap between the dispersion analysis and the actual error with source term. We illustrate it for classical and some dispersion free schemes in 1D, with conclusions verified by numerical experiments.