Energy-conserving semi-discretizations and spurious numerical reflections

We consider energy-conserving semi-discretizations of linear wave equations on nonuniform grids. Specifically we study explicit and implicit skew-adjoint finite difference methods, based on the assumption of an underlying smooth mapping from a uniform grid, applied to the first and second order wave equations. Our interest is in internal reflection of energy at abrupt variations in grid spacing. We show that all node-centered finite difference schemes suffer from reflections. Cell-centered finite difference schemes for the first order wave equation do not have reflections if the numerical dispersion relation is monotone. Runge-Kutta-based spatial semi-discretizations are also considered and these never give reflections. Furthermore, for higher order wave equations, even finite difference schemes with compact stencils and monotone dispersion relations may give reflections due to coupling of physically significant dispersion branches. Again RK schemes avoid this. Finally, we note that all schemes which avoid internal reflections are implicit