IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v347y2019icp194-209.html
   My bibliography  Save this article

Structure preserving reduced order modeling for gradient systems

Author

Listed:
  • Akman Yıldız, Tuğba
  • Uzunca, Murat
  • Karasözen, Bülent

Abstract

Minimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of ordinary differential equations (ODEs) are integrated in time by the average vector field (AVF) method, which preserves the energy dissipation of the gradient systems. The ROMs are constructed by the proper orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction terms are computed efficiently by discrete empirical interpolation method (DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM ensures the long term stability of the steady state solutions. Numerical simulations are performed for the gradient dissipative systems with two specific equations; real Ginzburg–Landau equation and Swift–Hohenberg equation. Numerical results demonstrate that the POD-DEIM reduced order solutions preserve well the energy dissipation over time and at the steady state.

Suggested Citation

  • Akman Yıldız, Tuğba & Uzunca, Murat & Karasözen, Bülent, 2019. "Structure preserving reduced order modeling for gradient systems," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 194-209.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:194-209
    DOI: 10.1016/j.amc.2018.11.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300318309810
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2018.11.008?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Bülent Karasözen & Tuğba Küçükseyhan & Murat Uzunca, 2017. "Structure preserving integration and model order reduction of skew-gradient reaction–diffusion systems," Annals of Operations Research, Springer, vol. 258(1), pages 79-106, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Xiaowei Chen & Mingzhan Song & Songhe Song, 2020. "A Fourth Order Energy Dissipative Scheme for a Traffic Flow Model," Mathematics, MDPI, vol. 8(8), pages 1-10, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Soheyl Khalilpourazari & Shima Soltanzadeh & Gerhard-Wilhelm Weber & Sankar Kumar Roy, 2020. "Designing an efficient blood supply chain network in crisis: neural learning, optimization and case study," Annals of Operations Research, Springer, vol. 289(1), pages 123-152, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:194-209. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.