A new Energy Equation Derivation for the Shallow Water Linearized Moment Equations
Julian Koellermeier
TL;DR
The paper addresses the problem of deriving a robust energy equation for the Shallow Water Linearized Moment Equations (SWLME). Building on the standard SWE energy derivation and adopting a skew-symmetric formulation, it performs a step-by-step energy analysis starting from the SWLME with $A_{ijk}=B_{ijk}=0$ to obtain a conserved total energy $e$ that includes both depth-averaged and higher-order vertical-velocity contributions via coefficients $u_i$. The main contributions are the explicit, systematic derivation of the SWLME energy equation, the explicit form of the energy $e$ and flux $f$ that incorporate the moment terms, and the identification of skew-symmetric structures in the momentum and moment equations. The findings facilitate energy-stable or entropy-conservative discretizations and provide a pathway to extend the methodology to other SWME variants such as Kowalski and dispersive models, with potential impact on robust numerical simulations of free-surface flows.
Abstract
Shallow Water Moment Equations (SWME) are extensions to the well-known Shallow Water Equations (SWE) for the efficient modeling and numerical simulation of free-surface flows. While the SWE typically assume a depth-averaged vertical velocity profile, the SWME allow for vertical variations of the velocity profile. The SWME therefore assume a polynomial profile and then derive additional evolution equations for the polynomial coefficients via higher order depth integration. In this work, we perform a new systematic derivation of the energy equation for a specific variant of the SWME, called the Shallow Water Linearized Moment Equations (SWLME). The derivation is based on the standard SWE energy equation derivation and includes the skew-symmetric formulation of the model. The new systematic derivation is beneficial for the extension to other SWME variants and their numerical solution.