The computation of nonclassical shock waves with a heterogeneous
multiscale method

Frederike Kissling; Christian Rohde; Frederike Kissling; Christian Rohde

doi:10.3934/nhm.2010.5.661

Networks and Heterogeneous Media

2010, Volume 5, Issue 3: 661-674. doi: 10.3934/nhm.2010.5.661

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The computation of nonclassical shock waves with a heterogeneous multiscale method

Frederike Kissling ^{1
,},
Christian Rohde ^{1
,}

1.
Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart

Received: 01 January 2010 Revised: 01 May 2010
Primary: 35L65.

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.
- Heterogeneous multiscale method,
- conservation laws,
- shock wave,
- nonclassical shock,
- phase transition,
- two-phase flow
Citation: Frederike Kissling, Christian Rohde. The computation of nonclassical shock waves with a heterogeneousmultiscale method[J]. Networks and Heterogeneous Media, 2010, 5(3): 661-674. doi: 10.3934/nhm.2010.5.661

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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.
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