Energy conservation and physical optics for discrete long wave equations

In a preceding paper [Wave Motion 32 (2000) 79] physical optics was developed for a selection of discrete long wave solutions by application a perturbation expansion. Herein, we approach discrete optics from analysis of discrete energy, in the sense of discrete expressions fulfilling a local energy conservation law. The expressions for energy density and flux are found from finite difference and element discretizations of linear and non-linear shallow water equations as well as linear Boussinesq equations with constant depth. The discrete energies are generally not perfectly positive definite and ambiguity in their definition is encountered and discussed. Still, the discrete energies appear as well behaved, which is confirmed through calculation of averaged energy quantities for harmonics, that reproduces the physical optics from Pedersen [Wave Motion 32 (2000) 79] under the assumption of negligible diffraction. The optics and the derived expressions for the energies are verified through direct solution of discrete long wave equations. Discrete energies are also discussed in relation to Hamilton's principle. The discrete momentum equation can be obtained from Hamilton's principle, inserted energy-like quadratures. However, those energy forms are not consistent with the energies that are conserved. (C) 2003 Elsevier Science B.V. All rights reserved.