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The shallow water equations provide a useful analogue of the fullycompressible Euler equations since they have similar characteristics:conservation laws, inertia-gravity and Rossby waves and a (quasi-) balancedstate. In order to obtain realistic simulation results, it is desirable thatnumerical models have discrete analogues of these properties. Two prototypicalexamples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid totalenergy and potential enstrophy conserving scheme, and the 2007 Salmon (S07)Z-grid total energy and potential enstrophy conserving scheme. Unfortunately,the AL81 scheme is restricted to logically square, orthogonal grids; and theS07 scheme is restricted to uniform square grids. The current work extends theAL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme toarbitrary orthogonal spherical polygonal grids in a manner that allows bothtotal energy and potential enstrophy conservation, by combining Hamiltonianmethods (work done by Salmon, Gassmann, Dubos and others) and Discrete ExteriorCalculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp and others).Detailed results of the schemes applied to standard test cases are deferred toPart 2 of this series of papers.

Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods: Derivation and Properties (Part 1)