An optical theory for discrete media

Abstract

This paper presents a new kind of analysis of numerical methods for water waves, where asymptotic techniques, previously associated with analytic theory only, are applied to discrete solutions. Observing the close relations between the asymptotic techniques and the basic concepts of wave theory, we realize that this approach directly access the reproduction of physical properties in the discrete description. Herein we focus on one theory of fundamental importance, namely that of geometrical and physical optics. We seek a corresponding description for waves that are discrete, in the sense of being solutions of difference rather than differential equations. Particularly, we address amplification of harmonic waves in shoaling water through an optical theory for discrete solutions, that is derived from a WKBJ type expansion. One of the key results is a discrete counterpart to Green’s law. We also find spurious reflection of short waves in shoaling water that can be described by a local expansion. Both the discrete Green’s law and the local approximation are verified by comparison to full, computed discrete solutions. Finally, corresponding asymptotic expansions are presented for a somewhat wider spectrum of methods, including finite element discretizations and a finite difference scheme for the linearized Boussinesq equations.