Riemann solvers for solving the incompressible Navier-Stokes equations using the artificial compressibility method

The solution to the Incompressible Navier-Stokes equations still represents a significant numerical challenge. The reason for this is that there is a lack of coupling between velocity and pressure. This means that the equations themselves provide no way of explicitly updating the pressure field as the velocity field is advanced. The artificial compressibility approach, devised by A. J. Chorin (see Chorin 1967), represents one way of overcoming this difficulty. It is arrived at by altering the incompressible equations in such a way as to result in a system of equations in which the left hand side is hyperbolic. We wish to take advantage of the hyperbolic nature of these equations and use Riemann-problem- based-numerical-methods (or RP methods).