  Solutions of initial-boundary value problems for systems of conservation laws
depend on the underlying viscous mechanism, namely different viscosity
operators lead to different limit solutions. Standard numerical schemes for
approximating conservation laws do not take into account this fact and converge
to solutions that are not necessarily physically relevant. We design numerical
schemes that incorporate explicit information about the underlying viscosity
mechanism and approximate the physically relevant solution. Numerical
experiments illustrating the robust performance of these schemes are presented.
