A relaxation method for steady Navier-Stokes equations, based on flux-vector splitting

Based on the example of Cauchy-Riemann equations, the flux-vector splitting method is illustrated for systems of first-order equations. It is shown that for this linear elliptic system of equations, flux-vector splitting combined with upwind differencing results in discrete equations which can be solved by relaxation methods. Furthermore, it is shown how the same splitting can be used on the hybrid first-order (subprincipal) part of the steady Navier-Stokes equations in a primitive variable form. By the use of central difference discretizations on the second order (principal) part, the resulting set of discrete equations can be solved by relaxation methods. A computational example of a backward-facing step problem is given.