IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v166y2023ics0960077922011080.html
   My bibliography  Save this article

On the nonstandard finite difference method for reaction–diffusion models

Author

Listed:
  • Pasha, Syed Ahmed
  • Nawaz, Yasir
  • Arif, Muhammad Shoaib

Abstract

The nonstandard finite difference (NSFD) method is an elegant approach in that it overcomes the numerical instability and bias exhibited by standard finite difference methods for numerically solving nonlinear differential equations. In addition, the NSFD method preserves some qualitative features of the continuous-time model such as boundedness and positivity. But for an important class of models which include diffusion and reaction–diffusion systems that appear in a number of application domains including epidemiology, ecology, and finance, recently introduced NSFD schemes do not guarantee first-order temporal accuracy or consistency. In this paper, we first show this for a reaction–diffusion epidemic model. We then propose an alternative NSFD scheme that guarantees first-order accuracy in time and second-order accuracy in space whilst preserving positivity of the solution. Stability and consistency analyses of the proposed scheme are then presented. To demonstrate the performance of the proposed scheme we show a comparison with the existing NSFD approach for three examples which confirm the superiority of our proposed approach.

Suggested Citation

  • Pasha, Syed Ahmed & Nawaz, Yasir & Arif, Muhammad Shoaib, 2023. "On the nonstandard finite difference method for reaction–diffusion models," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:chsofr:v:166:y:2023:i:c:s0960077922011080
    DOI: 10.1016/j.chaos.2022.112929
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077922011080
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2022.112929?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. Ahmed, Nauman & Rafiq, Muhammad & Adel, Waleed & Rezazadeh, Hadi & Khan, Ilyas & Nisar, Kottakkaran Sooppy, 2020. "Structure Preserving Numerical Analysis of HIV and CD4+T-Cells Reaction Diffusion Model in Two Space Dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    2. Elaiw, A.M. & Alshaikh, M.A., 2020. "Stability preserving NSFD scheme for a general virus dynamics model with antibody and cell-mediated responses," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    3. Company, Rafael & Jódar, Lucas & Pintos, José-Ramón, 2012. "A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(10), pages 1972-1985.
    4. Mickens, Ronald E., 2003. "A nonstandard finite difference scheme for the diffusionless Burgers equation with logistic reaction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(1), pages 117-124.
    5. Dimitrov, Dobromir T. & Kojouharov, Hristo V., 2008. "Nonstandard finite-difference methods for predator–prey models with general functional response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 78(1), pages 1-11.
    6. Wood, Daniel T. & Kojouharov, Hristo V. & Dimitrov, Dobromir T., 2017. "Universal approaches to approximate biological systems with nonstandard finite difference methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 133(C), pages 337-350.
    7. Mickens, Ronald E., 2016. "NSFD scheme for acoustic propagation with the linearized Euler equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 127(C), pages 189-193.
    8. Hongpeng Guo & Zhiming Guo, 2021. "Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation," Mathematics, MDPI, vol. 9(16), pages 1-20, August.
    9. Liu, Hong & Yong, Jiongmin, 2005. "Option pricing with an illiquid underlying asset market," Journal of Economic Dynamics and Control, Elsevier, vol. 29(12), pages 2125-2156, December.
    10. Ahmed, Nauman & Ali, Mubasher & Baleanu, Dumitru & Rafiq, Muhammad & Rehman, Muhammad Aziz ur, 2020. "Numerical analysis of diffusive susceptible-infected-recovered epidemic model in three space dimension," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    11. Jang, Junyoung & Kwon, Hee-Dae & Lee, Jeehyun, 2020. "Optimal control problem of an SIR reaction–diffusion model with inequality constraints," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 171(C), pages 136-151.
    12. Adamu, Elias M. & Patidar, Kailash C. & Ramanantoanina, Andriamihaja, 2021. "An unconditionally stable nonstandard finite difference method to solve a mathematical model describing Visceral Leishmaniasis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 171-190.
    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. Hoang, Manh Tuan, 2022. "Reliable approximations for a hepatitis B virus model by nonstandard numerical schemes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 32-56.
    2. Manuel L. Esquível & Nadezhda P. Krasii & Pedro P. Mota & Victoria V. Shamraeva, 2023. "Coupled Price–Volume Equity Models with Auto-Induced Regime Switching," Risks, MDPI, vol. 11(11), pages 1-20, November.
    3. Ahmadian, D. & Farkhondeh Rouz, O. & Ivaz, K. & Safdari-Vaighani, A., 2020. "Robust numerical algorithm to the European option with illiquid markets," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    4. Hoang, Manh Tuan, 2022. "Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 359-373.
    5. Olaf Korn & Paolo Krischak & Erik Theissen, 2019. "Illiquidity transmission from spot to futures markets," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(10), pages 1228-1249, October.
    6. Korkut, Sıla Ö. & Erdoğan, Utku, 2018. "Positivity preserving scheme based on exponential integrators," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 731-739.
    7. Abraham J. Arenas & Gilberto González-Parra & Jhon J. Naranjo & Myladis Cogollo & Nicolás De La Espriella, 2021. "Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay," Mathematics, MDPI, vol. 9(3), pages 1-21, January.
    8. Xingchun Wang, 2021. "Pricing vulnerable options with jump risk and liquidity risk," Review of Derivatives Research, Springer, vol. 24(3), pages 243-260, October.
    9. Bruno Bouchard & G. Loeper & Y. Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01611790, HAL.
    10. Li, Zhe & Zhang, Weiguo & Zhang, Yue & Yi, Zhigao, 2019. "An analytical approximation approach for pricing European options in a two-price economy," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    11. B Bouchard & G Loeper & Y Zou, 2015. "Hedging of covered options with linear market impact and gamma constraint," Papers 1512.07087, arXiv.org.
    12. Frédéric Abergel & Grégoire Loeper, 2013. "Pricing and hedging contingent claims with liquidity costs and market impact," Working Papers hal-00802402, HAL.
    13. Shih-Ping Feng, 2011. "The Liquidity Effect In Option Pricing: An Empirical Analysis," The International Journal of Business and Finance Research, The Institute for Business and Finance Research, vol. 5(2), pages 35-43.
    14. Sergey Isaenko & Rui Zhong, 2015. "Liquidity premium in the presence of stock market crises and background risk," Quantitative Finance, Taylor & Francis Journals, vol. 15(1), pages 79-90, January.
    15. Han, Lili & Song, Sha & Pan, Qiuhui & He, Mingfeng, 2023. "The impact of multiple population-wide testing and social distancing on the transmission of an infectious disease," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 630(C).
    16. Behzad Alimoradian & Karim Barigou & Anne Eyraud-Loisel, 2022. "Derivatives under market impact: Disentangling cost and information," Working Papers hal-03668432, HAL.
    17. Michail Anthropelos & Scott Robertson & Konstantinos Spiliopoulos, 2018. "Optimal Investment, Demand and Arbitrage under Price Impact," Papers 1804.09151, arXiv.org, revised Dec 2018.
    18. Youssef El-Khatib & Abdulnasser Hatemi-J, 2023. "On a regime switching illiquid high volatile prediction model for cryptocurrencies," Journal of Economic Studies, Emerald Group Publishing Limited, vol. 51(2), pages 485-498, July.
    19. Bruno Bouchard & G Loeper & Y Zou, 2016. "Almost-sure hedging with permanent price impact," Post-Print hal-01133223, HAL.
    20. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun, 2018. "Analytical valuation for geometric Asian options in illiquid markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 175-191.

    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:chsofr:v:166:y:2023:i:c:s0960077922011080. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.