Comparison and modification: TVD schemes for scalar transport on an unstructured grid

In most TVD schemes, the r-factors were proposed according to the cell-centered(CC) finite volume method(FVM) framework for the numerical approximation to the convective term. However, it is questionable whether those r-factors would be appropriate and effective for the vertex-centered(VC) FVM. In the paper, we collected five kinds of r-factor formulae and found out that only three of those, respectively by Bruner(1996), Darwish and Moukalled(2003) and Cassuli and Zanolli(2005) can be formally extended to a context of the VC FVM. Numerical tests indicate that the TVD schemes and r-factors, after being extended and introduced to a context of the VC FVM, maintained their similar characteristics as in a context of the CC FVM. However, when the gradient-based r-factors and the SUPERBEE scheme were applied simultaneously, non-physical oscillations near the sharp step would appear. In the transient case, the oscillations were weaker in a context of the VC FVM than those in a context of the CC FVM, while the effect was reversed in the steady case. To eliminate disadvantages in the gradient-based r-factor formula, a new modification method by limiting values on the virtual node, namely U? in the paper, was validated by the tests to effectively dissipate spurious oscillations.