# Variational integrators for anelastic and pseudo-incompressible flows

## TL;DR

This paper develops structure-preserving numerical schemes for anelastic and pseudo-incompressible flows using a variational approach based on discrete Lie groups, ensuring good energy behavior and applicability to irregular meshes.

## Contribution

It introduces a novel variational discretization method for these flow systems, preserving geometric structures and circulation properties.

## Key findings

- Schemes exhibit excellent long-term energy conservation.
- Applicable to irregular meshes with preserved divergence conditions.
- Preliminary tests demonstrate scheme stability and accuracy.

## Abstract

The anelastic and pseudo-incompressible equations are two well-known soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper, we derive and test structure-preserving numerical schemes for these two systems. The derivations are based on a discrete version of the Euler-Poincar\'e variational method. This approach relies on a finite dimensional approximation of the (Lie) group of diffeomorphisms that preserve weighted-volume forms. These weights describe the background stratification of the fluid and correspond to the weighed velocity fields for anelastic and pseudo-incompressible approximations. In particular, we identify to these discrete Lie group configurations the associated Lie algebras such that elements of the latter correspond to weighted velocity fields that satisfy the divergence-free conditions for both systems. Defining discrete Lagrangians in terms of these Lie algebras, the discrete equations follow by means of variational principles. Descending from variational principles, the schemes exhibit further a discrete version of Kelvin circulation theorem, are applicable to irregular meshes, and show excellent long term energy behavior. We illustrate the properties of the schemes by performing preliminary test cases.