# A direction splitting scheme for Navier-Stokes-Boussinesq system in spherical shell geometries

## TL;DR

This paper presents a second-order direction-splitting numerical scheme for solving the Navier-Stokes-Boussinesq system in spherical shells, utilizing overset Yin-Yang grids and parallel domain decomposition for improved stability and scalability.

## Contribution

It introduces a novel second-order direction-splitting method on overset Yin-Yang grids for spherical shell geometries, avoiding pole singularities and enabling parallel computation.

## Key findings

- Method achieves second-order temporal accuracy.
- Scheme demonstrates stability and weak scalability.
- Validated on manufactured and Landau solutions.

## Abstract

This paper introduces a formally second-order direction-splitting method for solving the incompressible Navier-Stokes-Boussinesq system in a spherical shell region. The equations are solved on overset Yin-Yang grids, combined with spherical coordinate transforms. This approach allows to avoid the singularities at the poles and keeps the grid size relatively uniform. The downside is that the spherical shell is subdivided into two equally sized, overlapping subdomains that requires the use of Schwarz-type iterations. The temporal second order accuracy is achieved via an Artificial Compressibility (AC) scheme with bootstrapping. The spatial discretization is based on second order finite differences on the Marker-And-Cell (MAC) stencil. The entire scheme is implemented in parallel using a domain decomposition iteration, and a direction splitting approach for the local solves. The stability, accuracy and weak scalability of the method is verified on a manufactured solution of the Navier-Stokes-Boussinesq system and on the Landau solution of the Navier-Stokes equations on the sphere.