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September  2010, 5(3): 661-674. doi: 10.3934/nhm.2010.5.661

The computation of nonclassical shock waves with a heterogeneous multiscale method

Frederike Kissling 1, and  Christian Rohde 1,

1. 

Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany, Germany

Received  January 2010 Revised  May 2010 Published  July 2010

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.
Keywords: nonclassical shock, phase transition, conservation laws, two-phase flow., Heterogeneous multiscale method, shock wave.
Mathematics Subject Classification: Primary: 35L6.
Citation: Frederike Kissling, Christian Rohde. The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks & Heterogeneous Media, 2010, 5 (3) : 661-674. doi: 10.3934/nhm.2010.5.661
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