A hyperbolic relaxation system of the incompressible Navier-Stokes equations with artificial compressibility

TL;DR

This paper presents a novel hyperbolic relaxation approach to approximate incompressible Navier-Stokes equations, utilizing artificial compressibility and relaxation parameters, with proven convergence to the classical equations as parameters vanish.

Contribution

It introduces a new hyperbolic relaxation system with two parameters for the incompressible Navier-Stokes equations and proves its asymptotic convergence to the classical model.

Findings

Proved asymptotic limit of the relaxation system to Navier-Stokes equations.

Established convergence of the approximate pressure variable.

Developed energy estimates for the error analysis.

Abstract

We introduce a new hyperbolic approximation to the incompressible Navier-Stokes equations by incorporating a first-order relaxation and using the artificial compressibility method. With two relaxation parameters in the model, we rigorously prove the asymptotic limit of the system towards the incompressible Navier-Stokes equations as both parameters tend to zero. Notably, the convergence of the approximate pressure variable is achieved by the help of a linear `auxiliary' system and energy-type error estimates of its differences with the two-parameter model and the Navier-Stokes equations.