IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v190y2021icp329-342.html
   My bibliography  Save this article

Accurate numerical method to solve flux distribution of Poisson’s equation

Author

Listed:
  • Hirokami, Arata
  • Heshmat, Samia
  • Tomioka, Satoshi

Abstract

This paper proposes an accurate numerical method, the direct flux method (DFM), to solve fluxes directly for Poisson’s equation. In DFM, fluxes are the variables to be solved in the system equations, where a flux is defined as an integral of the flux density across a certain finite-sized cross section. The system equation of the DFM is derived from two equations: an integral form of Poisson’s equation obtained by using Gauss’s divergence theorem and an integral form of the rotation-free nature of any scalar field from Stokes’ theorem. In the numerical approach, no discretization error arises from Gauss’s divergence theorem because it is represented as a sum of fluxes. Therefore, the discretization error is caused only by the integral form of the rotation-free nature. From the comparison between DFM, the finite difference method (FDM), and the finite volume method (FVM), we show that the accuracy of DFM is superior to that of FDM and FVM. However, DFM generally requires larger computational resources than other methods because the number of equations in DFM is more than that in other methods. To overcome this drawback, we also propose a faster algorithm than DFM, called FastDFM, which can reduce the number of equations without changing the accuracy. Hence, the proposed FastDFM produces results with the same accuracy as the DFM and with computation time almost the same as that of FDM and FVM.

Suggested Citation

  • Hirokami, Arata & Heshmat, Samia & Tomioka, Satoshi, 2021. "Accurate numerical method to solve flux distribution of Poisson’s equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 329-342.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:329-342
    DOI: 10.1016/j.matcom.2021.05.028
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475421001993
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2021.05.028?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. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(5), pages 687-698, October.
    2. Uh Zapata, Miguel & Itzá Balam, Reymundo, 2017. "High-order implicit finite difference schemes for the two-dimensional Poisson equation," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 222-244.
    3. Massimo Rundo, 2017. "Models for Flow Rate Simulation in Gear Pumps: A Review," Energies, MDPI, vol. 10(9), pages 1-32, August.
    4. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(3), pages 381-386, June.
    5. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(4), pages 525-537, August.
    6. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(2), pages 285-292, April.
    7. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 151-159, February.
    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. Talman, A.J.J. & Yang, Z.F., 2012. "On a parameterized system of nonlinear equations with economic applications," Other publications TiSEM 8233343d-0b60-428d-a20b-6, Tilburg University, School of Economics and Management.
    2. Zhiqiang Zheng & Balaji Padmanabhan & Steven O. Kimbrough, 2003. "On the Existence and Significance of Data Preprocessing Biases in Web-Usage Mining," INFORMS Journal on Computing, INFORMS, vol. 15(2), pages 148-170, May.
    3. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Discussion Paper 1999-72, Tilburg University, Center for Economic Research.
    4. Carlos R. Handy & Daniel Vrinceanu & Carl B. Marth & Harold A. Brooks, 2015. "Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure," Mathematics, MDPI, vol. 3(4), pages 1-24, November.
    5. Allen C. Goodman & Miron Stano, 2000. "Hmos and Health Externalities: A Local Public Good Perspective," Public Finance Review, , vol. 28(3), pages 247-269, May.
    6. Bode, Sven & Michaelowa, Axel, 2003. "Avoiding perverse effects of baseline and investment additionality determination in the case of renewable energy projects," Energy Policy, Elsevier, vol. 31(6), pages 505-517, May.
    7. Ala, Guido & Fasshauer, Gregory E. & Francomano, Elisa & Ganci, Salvatore & McCourt, Michael J., 2017. "An augmented MFS approach for brain activity reconstruction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 3-15.
    8. Bettina Campedelli & Andrea Guerrina & Giulia Romano & Chiara Leardini, 2014. "La performance della rete ospedaliera pubblica della regione Veneto. L?impatto delle variabili ambientali e operative sull?efficienza," MECOSAN, FrancoAngeli Editore, vol. 2014(92), pages 119-142.
    9. Haider A. Khan, 2004. "General Conclusions: From Crisis to a Global Political Economy of Freedom," Palgrave Macmillan Books, in: Global Markets and Financial Crises in Asia, chapter 9, pages 193-211, Palgrave Macmillan.
    10. Penn Loh & Zoë Ackerman & Joceline Fidalgo & Rebecca Tumposky, 2022. "Co-Education/Co-Research Partnership: A Critical Approach to Co-Learning between Dudley Street Neighborhood Initiative and Tufts University," Social Sciences, MDPI, vol. 11(2), pages 1-17, February.
    11. Broekhuis, Manda & Vos, Janita F.J., 2003. "Improving organizational sustainability using a quality perspective," Research Report 03A43, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    12. O'Brien, Raymond & Patacchini, Eleonora, 2003. "Testing the exogeneity assumption in panel data models with "non classical" disturbances," Discussion Paper Series In Economics And Econometrics 0302, Economics Division, School of Social Sciences, University of Southampton.
    13. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2002. "Perfection and Stability of Stationary Points with Applications in Noncooperative Games," Discussion Paper 2002-108, Tilburg University, Center for Economic Research.
    14. Edcarlos D. Silva & J. C. Albuquerque & T. R. Cavalcante, 2021. "Fourth-order nonlocal type elliptic problems with indefinite nonlinearities," Partial Differential Equations and Applications, Springer, vol. 2(2), pages 1-22, April.
    15. YongSeog Kim & W. Nick Street & Gary J. Russell & Filippo Menczer, 2005. "Customer Targeting: A Neural Network Approach Guided by Genetic Algorithms," Management Science, INFORMS, vol. 51(2), pages 264-276, February.
    16. Montijano, J.I. & Rández, L. & Van Daele, M. & Calvo, M., 2020. "On the numerical stability of the exponentially fitted methods for first order IVPs," Applied Mathematics and Computation, Elsevier, vol. 379(C).
    17. Yanling Li & Zita Oravecz & Shuai Zhou & Yosef Bodovski & Ian J. Barnett & Guangqing Chi & Yuan Zhou & Naomi P. Friedman & Scott I. Vrieze & Sy-Miin Chow, 2022. "Bayesian Forecasting with a Regime-Switching Zero-Inflated Multilevel Poisson Regression Model: An Application to Adolescent Alcohol Use with Spatial Covariates," Psychometrika, Springer;The Psychometric Society, vol. 87(2), pages 376-402, June.
    18. Jensen, Nathan M. & Li, Quan & Rahman, Aminur, 2007. "Heard melodies are sweet, but those unheard are sweeter : understanding corruption using cross-national firm-level surveys," Policy Research Working Paper Series 4413, The World Bank.
    19. Oscar J. Cacho & Robyn L. Hean & Russell M. Wise, 2003. "Carbon‐accounting methods and reforestation incentives," Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 47(2), pages 153-179, June.
    20. Walter M. Cadette, 1999. "Financing Long-Term Care: Options for Policy," Economics Working Paper Archive wp_283, Levy Economics Institute.

    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:190:y:2021:i:c:p:329-342. 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.