# A linear and nonlinear analysis of the shallow water equations and its impact on boundary conditions

## TL;DR

This paper compares linear and nonlinear analyses of the shallow water equations, revealing differences in boundary condition requirements and emphasizing the importance of nonlinear energy methods for accurate modeling.

## Contribution

It introduces a nonlinear energy analysis approach that refines boundary condition estimates for the shallow water equations, highlighting differences from linear analysis.

## Key findings

- Nonlinear analysis alters the minimal number of boundary conditions needed.
- Flow magnitude does not affect boundary condition requirements in the nonlinear case.
- Energy and entropy analyses are consistent but provide different insights.

## Abstract

We derive boundary conditions and estimates based on the energy and entropy analysis of systems of the nonlinear shallow water equations in two spatial dimensions. It is shown that the energy method provides more details, but is fully consistent with the entropy analysis. The details brought forward by the nonlinear energy analysis allow us to pinpoint where the difference between the linear and nonlinear analysis originate. We find that the result from the linear analysis does not necessarily hold in the nonlinear case. The nonlinear analysis leads in general to a different minimal number of boundary conditions compared with the linear analysis. In particular, and contrary to the linear case, the magnitude of the flow does not influence the number of required boundary conditions.