IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v208y2023icp71-94.html
   My bibliography  Save this article

Proper Generalized Decomposition using Taylor expansion for non-linear diffusion equations

Author

Listed:
  • Deeb, Ahmad
  • Kalaoun, Omar
  • Belarbi, Rafik

Abstract

For a physical problem described by a parameterized mathematical model, different configurations of the problem require computing the solution over a range of parameters in order to study the phenomenon when parameters change. In other words, it is a process of looking for a continuum of solutions of the equation, relative to these parameters, in order to find the ones that fit the experimental data. However, solving a direct problem for each parametric configuration will generate a cascade of direct problems, which will cost a huge amount of time, especially when we deal with non-linear equations. Therefore, the parametric solution is a suitable alternative strategy to compute the solution of the equation. In this paper, we will use the Proper Generalized Decomposition (PGD) method to solve non-linear diffusion equations and produce parametric solutions. To treat the non-linear functions, we will not use the Discrete Empirical Interpolation Methods (DEIM), which has proven their utility, but the non-linear terms will be replaced by their Taylor series expansion up to an order m. This will produce a new model, which we call here the ”developed equation” and therefore the PGD is applied on. Polynomial equations appear for each tensor element computation. While space and time tensor elements’ equations are to be solved using Finite Elements Methods (FEM) and Borel–Padé–Laplace (BPL) integrator respectively, Newton solver is used for tensors relative to the parameters’ equations. Here, rational polynomial functions arise for parametric tensor elements, which are known to extrapolate solutions. Numerical simulations are done for a non-linear diffusion equation with exponential diffusion coefficient as first trial, and with a magnetic diffusion coefficient as a second one.

Suggested Citation

  • Deeb, Ahmad & Kalaoun, Omar & Belarbi, Rafik, 2023. "Proper Generalized Decomposition using Taylor expansion for non-linear diffusion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 71-94.
  • Handle: RePEc:eee:matcom:v:208:y:2023:i:c:p:71-94
    DOI: 10.1016/j.matcom.2023.01.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475423000071
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.matcom.2023.01.008?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. Deeb, Ahmad & Hamdouni, Aziz & Razafindralandy, Dina, 2022. "Performance of Borel–Padé–Laplace integrator for the solution of stiff and non-stiff problems," Applied Mathematics and Computation, Elsevier, vol. 426(C).
    2. Theeraek, P. & Phongthanapanich, S. & Dechaumphai, P., 2011. "Solving convection-diffusion-reaction equation by adaptive finite volume element method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(2), pages 220-233.
    3. Berger, Julien & Mendes, Nathan, 2017. "An innovative method for the design of high energy performance building envelopes," Applied Energy, Elsevier, vol. 190(C), pages 266-277.
    4. Gambino, G. & Lombardo, M.C. & Sammartino, M., 2012. "Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(6), pages 1112-1132.
    5. Amin, Fahs & Zakeri, Ali & Wanko, Adrien, 2021. "Time-dependent solution for natural convection in a porous enclosure using the Darcy–Lapwood–Brinkman model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 39-65.
    Full references (including those not matched with items on IDEAS)

    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. Christian Kuehn & Cinzia Soresina, 2020. "Numerical continuation for a fast-reaction system and its cross-diffusion limit," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-26, April.
    2. Karasözen, Bülent & Mülayim, Gülden & Uzunca, Murat & Yıldız, Süleyman, 2021. "Reduced order modelling of nonlinear cross-diffusion systems," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    3. Wang, Fatao & Yang, Ruizhi, 2023. "Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    4. Ghorai, Santu & Poria, Swarup, 2016. "Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 421-429.
    5. Flor, Jan-Frederik & Liu, Dingming & Sun, Yanyi & Beccarelli, Paolo & Chilton, John & Wu, Yupeng, 2018. "Optical aspects and energy performance of switchable ethylene-tetrafluoroethylene (ETFE) foil cushions," Applied Energy, Elsevier, vol. 229(C), pages 335-351.
    6. Li, Qiang & Liu, Zhijun & Yuan, Sanling, 2019. "Cross-diffusion induced Turing instability for a competition model with saturation effect," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 64-77.
    7. O'Grady, Małgorzata & Lechowska, Agnieszka A. & Harte, Annette M., 2017. "Quantification of heat losses through building envelope thermal bridges influenced by wind velocity using the outdoor infrared thermography technique," Applied Energy, Elsevier, vol. 208(C), pages 1038-1052.
    8. Wang, Fatao & Yang, Ruizhi & Zhang, Xin, 2024. "Turing patterns in a predator–prey model with double Allee effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 170-191.
    9. Zhang, Feifan & Sun, Jiamin & Tian, Wang, 2022. "Spatiotemporal pattern selection in a nontoxic-phytoplankton - toxic-phytoplankton - zooplankton model with toxin avoidance effects," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    10. Peng, Yahong & Zhang, Tonghua, 2016. "Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 1-12.
    11. Banda, Heather & Chapwanya, Michael & Dumani, Phindile, 2022. "Pattern formation in the Holling–Tanner predator–prey model with predator-taxis. A nonstandard finite difference approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 336-353.
    12. Mohan, Nishith & Kumari, Nitu, 2021. "Positive steady states of a SI epidemic model with cross diffusion," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    13. Lili Kusumawati & Erni Setyowati & Agus Budi Purnomo, 2021. "Practical-Empirical Modeling on Envelope Design towards Sustainability in Tropical Architecture," Sustainability, MDPI, vol. 13(5), pages 1-23, March.

    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:matcom:v:208:y:2023:i:c:p:71-94. 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: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    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.