Improved long time accuracy for projection methods for Navier-Stokes equations using EMAC formulation

TL;DR

This paper introduces an EMAC-based pressure correction method for the Navier-Stokes equations that improves long-term accuracy by reducing the exponential dependence on the Reynolds number, with proven stability and momentum conservation.

Contribution

It provides the first stability and error analysis for EMAC formulation with pressure correction, demonstrating reduced Gronwall constant dependence and momentum preservation.

Findings

Reduces exponential Gronwall dependence on Reynolds number.

Preserves momentum and angular momentum.

Numerical tests show advantages over other formulations.

Abstract

We consider a pressure correction temporal discretization for the incompressible Navier-Stokes equations in EMAC form. We prove stability and error estimates for the case of mixed finite element spatial discretization, and in particular that the Gronwall constant's exponential dependence on the Reynolds number is removed (for sufficiently smooth true solutions) or at least significantly reduced compared to the commonly used skew-symmetric formulation. We also show the method preserves momentum and angular momentum, and while it does not preserve energy it does admit an energy inequality. Several numerical tests show the advantages EMAC can have over other commonly used formulations of the nonlinearity. Additionally, we discuss extensions of the results to the usual Crank-Nicolson temporal discretization.