The computation of nonclassical shock waves with a heterogeneous multiscale method

Abstract

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 microscale model. With this point of view it appears natural to solve the macroscale model by a heterogeneous multiscale approach in the sense of E&Engquist[7]. We introduce a new mass-conserving numerical method based on this concept and test it on scalar model problems. This includes applications from phase transition theory as well as from two-phase flow in porous media. © American Institute of Mathematical Sciences.