  A high-order Flux reconstruction implementation of the hyperbolic formulation
for the incompressible Navier-Stokes equation is presented. The governing
equations employ Chorin's classical artificial compressibility (AC) formulation
cast in hyperbolic form. Instead of splitting the second-order conservation law
into two equations, one for the solution and another for the gradient, the
Navier-Stokes equation is cast into a first-order hyperbolic system of
equations. Including the gradients in the AC iterative process results in a
significant improvement in accuracy for the pressure, velocity, and its
gradients. Furthermore, this treatment allows for taking larger time-steps
since the hyperbolic formulation eliminates the restriction due to diffusion.
Tests using the method of manufactured solutions show that solving the
conventional form of the Navier-Stokes equation lowers the order of accuracy
for gradients, while the hyperbolic method is shown to provide equal orders of
accuracy for both the velocity and its gradients which may be beneficial in
several applications. Two- and three-dimensional benchmark tests demonstrate
the superior accuracy and computational efficiency of the developed solver in
comparison to the conventional method and other published works. This study
shows that the developed high-order hyperbolic solver for incompressible flows
is attractive due to its accuracy, stability and efficiency in solving
diffusion dominated problems.
