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Energy Stable Model Order Reduction for the Allen-Cahn Equation

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Model Reduction of Parametrized Systems

Part of the book series: MS&A ((MS&A,volume 17))

Abstract

The Allen-Cahn equation is a gradient system, where the free-energy functional decreases monotonically in time. We develop an energy stable reduced order model (ROM) for a gradient system, which inherits the energy decreasing property of the full order model (FOM). For the space discretization we apply a discontinuous Galerkin (dG) method and for time discretization the energy stable average vector field (AVF) method. We construct ROMs with proper orthogonal decomposition (POD)-greedy adaptive sampling of the snapshots in time and evaluating the nonlinear function with greedy discrete empirical interpolation method (DEIM). The computational efficiency and accuracy of the reduced solutions are demonstrated numerically for the parametrized Allen-Cahn equation with Neumann and periodic boundary conditions.

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Acknowledgements

The authors would like to thank the reviewer for the comments and suggestions that helped to improve the manuscript.

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Turkish Aeronautical Association, Ankara, Turkey

    Murat Uzunca

  2. Institute of Applied Mathematics and Department of Mathematics, Middle East Technical University, Ankara, Turkey

    Bülent Karasözen

Authors
  1. Murat Uzunca
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  2. Bülent Karasözen
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Correspondence to Bülent Karasözen .

Editor information

Editors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Institute for Computational and Applied Mathematics, University of Münster , Münster, Germany

    Mario Ohlberger

  3. Massachusetts Institute of Technology, MIT , Cambridge, Massachusetts, USA

    Anthony Patera

  4. SISSA MathLab, International School for Adv. Studies , Trieste, Italy

    Gianluigi Rozza

  5. Institute of Numerical Mathematics, Ulm University , Ulm, Germany

    Karsten Urban

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Uzunca, M., Karasözen, B. (2017). Energy Stable Model Order Reduction for the Allen-Cahn Equation. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_25

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