The computation of nonclassical shock waves with a heterogeneous multiscale method

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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

Abstract
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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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