Choice of implicit and explicit operators for the upwind differencing method

The flux-vector and flux-difference splittings of Steger-Warming, Van Leer and Roe are tested in all possible combinations in the implicit and explicit operators that can be distinguished in implicit relaxation methods for the steady Euler and Navier-Stokes equations. The tests include one-dimensional inviscid nozzle flow, and two-dimensional inviscid and viscous shock reflection. Roe's splitting, as anticipated, is found to uniformly yield the most accurate results. On the other hand, an approximate Roe splitting of the implicit operator (the complete Roe splitting is too complicated for practical use) proves to be the least robust with regard to convergence to the steady state. In this respect, the Steger-Warming splitting is the most robust: it leads to convergence when combined with any of the splittings in the explicit operator, although not necessarily in the most efficient way.