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

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.