# Symmetry-preserving finite-difference discretizations of arbitrary order on structured curvilinear staggered grids

## TL;DR

This paper introduces a new finite-difference discretization method that preserves symmetry and conservation properties on structured curvilinear grids, supporting arbitrary order and complex operators, demonstrated through shallow-water equations.

## Contribution

It presents a novel symmetry-preserving finite-difference scheme compatible with arbitrary order, various grid types, and complex operators, expanding the applicability of mimetic discretizations.

## Key findings

- Exact conservation of physical quantities demonstrated
- Convergence matches theoretical order
- Applicable to complex operators like chain rules and nonlinear advection

## 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. This paper presents a new finite-difference symmetry-preserving space discretization. Boundary conditions and time integration are not addressed. The novelty is that it combines arbitrary order of convergence, orthogonal and non-orthogonal structured curvilinear staggered meshes, and the applicability to a wide variety of continuous operators, involving chain rules and nonlinear advection, as illustrated by the shallow-water equations. Experiments show exact conservation and convergence corresponding to expected order.