An EMA-conserving, pressure-robust and Re-semi-robust reconstruction method for the unsteady incompressible Navier-Stokes equations

TL;DR

This paper introduces a new reconstruction method for incompressible Navier-Stokes simulations that preserves key physical properties like energy, momentum, and pressure-robustness, improving accuracy across different Reynolds numbers.

Contribution

It proposes a novel EMA-conserving, pressure-robust, and Re-semi-robust reconstruction method for non-divergence-free elements, extending pressure-robust techniques to broader element classes.

Findings

The method conserves kinetic energy, momentum, and angular momentum under certain conditions.

Numerical tests confirm improved accuracy and robustness compared to existing methods.

The approach effectively handles high Reynolds number flows with reduced error dependence on viscosity.

Abstract

Proper EMA-balance (E: kinetic energy; M: momentum; A: angular momentum), pressure-robustness and $Re$-semi-robustness ($Re$: Reynolds number) are three important properties of Navier-Stokes simulations with exactly divergence-free elements. This EMA-balance makes a method conserve kinetic energy, linear momentum and angular momentum under some suitable senses; pressure-robustness means that the velocity errors are independent of the continuous pressure; $Re$-semi-robustness means that the constants appearing in the error bounds of kinetic and dissipation energies do not explicitly depend on inverse powers of the viscosity. In this paper, based on the pressure-robust reconstruction methods in [{A. Linke and C. Merdon, {\it Comput. Methods Appl. Mech. Engrg.} 311 (2016), 304-326}], we propose a novel reconstruction method for a class of non-divergence-free simplicial elements which admits almost all the above properties. The only exception is the energy balance, where kinetic energy should be replaced by a properly redefined discrete energy. Some numerical comparisons with exactly divergence-free methods, pressure-robust reconstructions and the EMAC scheme are provided to confirm our theoretical results.