Many modern shock capturing schemes can be
classified as characteristic based schemes. These
rely on a local diagonalization of the hyperbolic
system that involves the Jacobian matrix of the flux
function at each cell interface. It is now
relatively well known that the interface state used
for the computation of the Jacobian can have a
significant effect on the fine features of the
numerical solution . A new numerical flux
function is developed in  that takes into account
the two physically relevant states at the interface,
without ever constructing an artificially mixed
interface state. This new flux function reduces to
the well known Shu-Osher scheme  for scalar
conservation laws, but becomes fundamentally
different in the case of systems. The resulting
numerical scheme can be combined with the ENO-reconstruction
technology to produce state of the art HRSC schemes
that have been applied in various scenarios. This
talk will review recent results that derive from the
use of these new schemes, with special emphasis in
the advantages and shortcomings found in the
 R. P. Fedkiw, B. Merriman, R.
Donat, S. Osher, The Penultimate Scheme for Systems
Conservation Laws: Finite Difference ENO with
Marquina's Flux Splitting, Innovative methods for
Numerical solutions of PDEs, Ed. M. M. Hafez, J. J.
Chattot 49, 1998.
 R. Donat and A. Marquina, Capturing shock
reflections: An improved flux formula, J. Comp.
Phys., 125, 1996.
 C. W Shu and S. J. Osher, Efficient
implementation of essentially non-oscillatory,
shock-capturing schemes II. J. Comp. Phys.,
*With G. Chiavassa and G. Haro