We consider weak solutions of hyperbolic conservation laws as
singular limits of solutions for associated complex regularized
problems. We are interested in situations such that
undercompressive (Non-Laxian) shock waves occur in the limit. In
this setting one can view the conservation law as a macroscale
formulation while the regularization can be understood as the
With this point of view it appears natural
to solve the macroscale model by a heterogeneous multiscale
approach in the sense of E&Engquist.
We introduce a new mass-conserving numerical method based on this
and test it on scalar model problems.
This includes applications from phase transition
theory as well as from two-phase flow in porous media.