IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i9p924-d540563.html
   My bibliography  Save this article

Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD

Author

Listed:
  • Marzieh Raei

    (Department of Applied Mathematics, Malek Ashtar University of Technology, Tehran 158751774, Iran)

  • Salvatore Cuomo

    (Scuola Politecnica e delle Scienze di Base, University of Naples Federico II, 80138 Napoli, Italy)

  • Giovanni Colecchia

    (Scuola Politecnica e delle Scienze di Base, University of Naples Federico II, 80138 Napoli, Italy)

  • Gerardo Severino

    (Scuola di Agraria e Medicina Veterinaria, University of Naples Federico II, 80138 Napoli, Italy)

Abstract

The Gray–Scott (GS) model is a non-linear system of equations generally adopted to describe reaction–diffusion dynamics. In this paper, we discuss a numerical scheme for solving the GS system. The diffusion coefficients of the model are on surfaces and they depend on space and time. In this regard, we first adopt an implicit difference stepping method to semi-discretize the model in the time direction. Then, we implement a hybrid advanced meshless method for model discretization. In this way, we solve the GS problem with a radial basis function–finite difference (RBF-FD) algorithm combined with the closest point method (CPM). Moreover, we design a predictor–corrector algorithm to deal with the non-linear terms of this dynamic. In a practical example, we show the spot and stripe patterns with a given initial condition. Finally, we experimentally prove that the presented method provides benefits in terms of accuracy and performance for the GS system’s numerical solution.

Suggested Citation

  • Marzieh Raei & Salvatore Cuomo & Giovanni Colecchia & Gerardo Severino, 2021. "Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD," Mathematics, MDPI, vol. 9(9), pages 1-20, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:924-:d:540563
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/9/924/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/9/924/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Gerardo Severino & Alessandro Santini & Valeria Marina Monetti, 2009. "Modelling Water Flow water flow and Solute Transport in Heterogeneous Unsaturated Porous Media," Springer Optimization and Its Applications, in: Panos M. Pardalos & Petraq J. Papajorgji (ed.), Advances in Modeling Agricultural Systems, pages 361-383, Springer.
    2. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(5), pages 777-788, October.
    3. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(1), pages 151-160, February.
    4. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(4), pages 629-637, August.
    5. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(3), pages 427-432, June.
    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. Krzysztof S. Targiel & Maciej Nowak & Tadeusz Trzaskalik, 2018. "Scheduling non-critical activities using multicriteria approach," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(3), pages 585-598, September.
    2. F. Castro-Llanos & G. Hyman & J. Rubiano & J. Ramirez-Villegas & H. Achicanoy, 2019. "Climate change favors rice production at higher elevations in Colombia," Mitigation and Adaptation Strategies for Global Change, Springer, vol. 24(8), pages 1401-1430, December.
    3. Okitonyumbe Y.F., Joseph & Ulungu, Berthold E.-L., 2013. "Nouvelle caractérisation des solutions efficaces des problèmes d’optimisation combinatoire multi-objectif [New characterization of efficient solution in multi-objective combinatorial optimization]," MPRA Paper 66123, University Library of Munich, Germany.
    4. Amit Kumar & Anila Gupta, 2013. "Mehar’s methods for fuzzy assignment problems with restrictions," Fuzzy Information and Engineering, Springer, vol. 5(1), pages 27-44, March.
    5. Monica Motta & Caterina Sartori, 2020. "Normality and Nondegeneracy of the Maximum Principle in Optimal Impulsive Control Under State Constraints," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 44-71, April.
    6. Zhang, Quanzhong & Wei, Haiyan & Liu, Jing & Zhao, Zefang & Ran, Qiao & Gu, Wei, 2021. "A Bayesian network with fuzzy mathematics for species habitat suitability analysis: A case with limited Angelica sinensis (Oliv.) Diels data," Ecological Modelling, Elsevier, vol. 450(C).
    7. Chenchen Wu & Dachuan Xu & Donglei Du & Wenqing Xu, 2016. "An approximation algorithm for the balanced Max-3-Uncut problem using complex semidefinite programming rounding," Journal of Combinatorial Optimization, Springer, vol. 32(4), pages 1017-1035, November.
    8. Gengping Zhu & Matthew J Petersen & Wenjun Bu, 2012. "Selecting Biological Meaningful Environmental Dimensions of Low Discrepancy among Ranges to Predict Potential Distribution of Bean Plataspid Invasion," PLOS ONE, Public Library of Science, vol. 7(9), pages 1-9, September.
    9. Uzma Ashraf & Hassan Ali & Muhammad Nawaz Chaudry & Irfan Ashraf & Adila Batool & Zafeer Saqib, 2016. "Predicting the Potential Distribution of Olea ferruginea in Pakistan incorporating Climate Change by Using Maxent Model," Sustainability, MDPI, vol. 8(8), pages 1-11, July.
    10. Ernst Althaus & Felix Rauterberg & Sarah Ziegler, 2020. "Computing Euclidean Steiner trees over segments," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(3), pages 309-325, October.
    11. World Bank, 2003. "Argentina : Reforming Policies and Institutions for Efficiency and Equity of Public Expenditures," World Bank Publications - Reports 14637, The World Bank Group.
    12. Ceretani, Andrea N. & Salva, Natalia N. & Tarzia, Domingo A., 2018. "Approximation of the modified error function," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 607-617.
    13. Parihar, Amit Kumar Singh & Hammer, Thomas & Sridhar, G., 2015. "Development and testing of tube type wet ESP for the removal of particulate matter and tar from producer gas," Renewable Energy, Elsevier, vol. 74(C), pages 875-883.
    14. Liang, Wanwan & Papeş, Monica & Tran, Liem & Grant, Jerome & Washington-Allen, Robert & Stewart, Scott & Wiggins, Gregory, 2018. "The effect of pseudo-absence selection method on transferability of species distribution models in the context of non-adaptive niche shift," Ecological Modelling, Elsevier, vol. 388(C), pages 1-9.
    15. Brown, Jeffrey R., 2001. "Private pensions, mortality risk, and the decision to annuitize," Journal of Public Economics, Elsevier, vol. 82(1), pages 29-62, October.
    16. Mark Christensen, 2007. "What We Might Know (But Aren't Sure) About Public-Sector Accrual Accounting," Australian Accounting Review, CPA Australia, vol. 17(41), pages 51-65, March.
    17. Wong, Patricia J.Y., 2015. "Eigenvalues of a general class of boundary value problem with derivative-dependent nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 908-930.
    18. Norma M Rantisi & Deborah Leslie, 2021. "In and against the neoliberal state? The precarious siting of work integration social enterprises (WISEs) as counter-movement in Montreal, Quebec," Environment and Planning A, , vol. 53(2), pages 349-370, March.
    19. Brunekreeft, Gert, 2004. "Market-based investment in electricity transmission networks: controllable flow," Utilities Policy, Elsevier, vol. 12(4), pages 269-281, December.
    20. Christophe Botella & Alexis Joly & Pascal Monestiez & Pierre Bonnet & François Munoz, 2020. "Bias in presence-only niche models related to sampling effort and species niches: Lessons for background point selection," PLOS ONE, Public Library of Science, vol. 15(5), pages 1-18, May.

    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:gam:jmathe:v:9:y:2021:i:9:p:924-:d:540563. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.