Any solver for hyperbolic equations can be applied to the scala advection equation written as \phi_t +( u \phi)_x =0. However, most standard schemes reduce to some combination of the upwind and Lax Wendroff methods for this simple case. Furthermore, many specific features of HLLC that are important for the Euler equations become irrelevant for linear advection, so that even applying an apparently more sophisticated method one should keep in mind that the result will be that of a basic second order (at most) solver. Should this introduce excessive numerical dissipation in your simulation, higher order advection schemes may have to be considered.

I agree, the linear advection equation can be used to test only some of the numerical methods features. In particular, the AUSM method is based on the splitting between convective and pressure waves.

I suggest to consider at least the Burgers equation to test some features for non linear equations.

I'm interested to couple linear advection equation with other conservative equations, such in the case of turbulence of multihpase.

The upwinding in HLLC for linear advection equation is according to the star wave, which is equivalent to the contact wave.  In AUSM the upwinding is according to the face velocity, and the physical meaning is not clear or how to derive a simpler equation for the upwinding in this case.

Well,  in turbulence you have to solve the equations for non-linear and viscous flows, while schemes such as AUSM are more focused on compressible inviscid flows.

Then, the discretization of the convective term depends on the formulation you adopt for simulating turbulence. DNS, LES, RANS requires quite different discretizations.

In the context of finite volume, I treat viscous terms using central scheme according to divergence theorem, the inviscid part is solved either through Roe/HLLC/AUSM.

In k-w RANS formulation, there are two additional transport equations for the two turbulence scales, while all other terms considered as source term.

Using RANS, the adopted numerical discretization never reaches a high order of accuracy since the turbulence model has a magnitude quite greater than the local truncation error.

In compressible flows, you will typically not solve for the turbulent variable, eg. k, but rather for $\rho k$. A good option is then to use the numerical/stabilised mass flow, ie. the one computed by your flux scheme for the Euler equations (here AUSM). This is, if you  the semidiscrete system for mass reads for a control volume V

$\int_V \fraction{\partial \rho}{\partial t} \ho dV + \sum_f \int_f (\rho u_n)^\ast dS = 0$

with f running over the faces of the control volume V and $(\rho u_n)^\ast$ is the mass flux through face f given by the flux formulation (AUSM in your case). Then you discretise the (conservative) advection of k

\int_V \fraction{\partial \rho k}{\partial t} + \sum_f \int_f (\rho u_n)^\ast k^\ast dS

where $k^\ast$ can be chosen on the basis of simple upwinding based upon the sign of $(\rho u_n)^\ast$. Using this you ensure a constant $k$ will remain exactly constant.

Koen Hillewaert,

If I follow simple upwinding, M_{1/2} serves as the upwinding velcoity for the density,

and therefore can be considered as the contact wave velocity compared with HLLC solver?

I follow the notation in AUSM-+up paper: "A sequel to AUSM, Part II: AUSM+-up for all speeds", see Eq. 9.

Thanks.

not exactly; you take the mass flux through face f resulting from the ausm flux, ie the discrete equivalent of $\rho v_n$ you use in your mass conservation equation, and you use it to upwind k (not $\rho k$).