# Derivations of continuous and discrete energy equations in wave and shallow-water equations

## TL;DR

This paper provides a detailed derivation of energy equations in symmetry-preserving discretizations of wave and shallow-water equations, building on previous work that ensured mass and momentum conservation.

## Contribution

It offers a comprehensive derivation of energy equations for mimetic discretizations, enhancing understanding of energy conservation in these models.

## Key findings

- Energy equations derived in detail for discrete models
- Conservation of mass and momentum established in previous work
- Enhanced theoretical foundation for stability and conservation properties

## Abstract

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.