A realizable, objective second-moment turbulence closure, allowing for an entropy caracterisation, is analyzed with respect to its convective subset. The distinct characteristic wave system of these equations in non-conservation form is exposed. An approximate solution to the associated one-dimensional Riemann problem is constructed making use of approximate jump conditions obtained by assuming a linear path across shock waves. A numerical integration method based on a new approximate Riemann solver (flux-difference-splitting) is proposed for use in conjunction with either unstructured or structured grids. Test calculations of quasi one-dimensional flow cases demonstrate the feasibility of the current technique even where Euler-based approaches fail.