Derivations of continuous and discrete energy equations in wave and shallow-water equations
For researchers working on mimetic discretizations, this paper fills a gap by providing complete energy conservation derivations, but it is an incremental extension of prior work.
This paper provides detailed derivations of energy equations for symmetry-preserving discretizations of wave and shallow-water equations, which were previously omitted due to space constraints. The derivations confirm exact energy conservation in the discrete models, matching the continuous case.
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and energy are proven in the same way as for the original continuous model. In our papers arXiv:1710.07149 and arXiv:1901.02264, we presented space discretization schemes for various models, which had exact conservation of mass, momentum and energy. Mass and momentum conservation followed from the left null spaces of the discrete operators used. The conservation of energy in the continuous and discrete models is more complicated, and the papers had little space for their complete derivation. This paper contains the derivation of the energy equations in more detail than was given in the papers arXiv:1710.07149 and arXiv:1901.02264.