# Well-balanced mesh-based and meshless schemes for the shallow-water equations

## TL;DR

This paper develops a unified approach for designing mesh-based and meshless numerical schemes that exactly preserve the lake at rest state in shallow-water equations, ensuring stability and accuracy.

## Contribution

It introduces a general criterion for mimetic design of derivative operators that guarantees the well-balanced property in both mesh-based and meshless schemes.

## Key findings

- Proves the consistency of mimetic difference operators analytically.
- Demonstrates the well-balanced property numerically in 1D and 2D cases.
- Applies the approach to finite difference and RBF-FD schemes.

## Abstract

We formulate a general criterion for the exact preservation of the "lake at rest" solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. The main idea is a careful mimetic design for the spatial derivative operators in the momentum flux equation that is paired with a compatible averaging rule for the water column height arising in the bottom topography source term. We prove consistency of the mimetic difference operators analytically and demonstrate the well-balanced property numerically using finite difference and RBF-FD schemes in the one- and two-dimensional cases.