# On exactly incompressible DG FEM pressure splitting schemes for the Navier-Stokes equation

## TL;DR

This paper compares three pressure correction schemes for Navier-Stokes equations using high-order discontinuous Galerkin methods, highlighting their divergence-free properties and computational efficiency.

## Contribution

It provides a detailed comparison of IPCS-D, IPCS-A, and SIMPLE schemes, demonstrating the algebraic pressure correction's advantage in mass conservation and efficiency.

## Key findings

- IPCS-A and SIMPLE are exactly mass conserving.
- IPCS-D cannot be exactly divergence free due to stabilization.
- IPCS-A is more efficient than SIMPLE in convergence.

## Abstract

We compare three iterative pressure correction schemes for solving the Navier-Stokes equations with a focus on exactly divergence free solution with higher order discontinuous Galerkin discretisations. The investigated schemes are the incremental pressure correction scheme on the standard differential form (IPCS-D), the same scheme on algebraic form (IPCS-A), and the semi-implicit method for pressure linked equations (SIMPLE). We show algebraically and through numerical examples that the IPCS-A and SIMPLE schemes are exactly mass conserving due to the algebraic pressure correction, while the IPCS-D scheme cannot be exactly divergence free due to the stabilisation terms required in the pressure Poisson equation. The SIMPLE scheme requires a significantly higher number of pressure correction iterations to obtain converged results than the IPCS-A scheme, so for efficient and mass conserving simulation the IPCS-A method is the best option among the three evaluated schemes.