IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v307y2017icp329-341.html
   My bibliography  Save this article

An efficient parareal algorithm for a class of time-dependent problems with fractional Laplacian

Author

Listed:
  • Wu, Shulin

Abstract

Time-dependent diffusion equations with fractional Laplacian have received considerable attention in recent years, for which numerical methods play an important role because a simple and analytic solution is often unavailable. We analyze in this paper a parareal algorithm for this kind of problem, which realizes parallel-in-time computation. The algorithm is iterative and uses the 3rd-order SDIRK (singly diagonally implicit Runge-Kutta) method with a small step-size Δt as the F-propagator and the implicit-explicit Euler method with a large step-size ΔT as the G-propagator. The two step-sizes satisfy ΔT/Δt=J with J ≥ 2 being an integer. Using the implicit-explicit Euler method as the G-propagator potentially improves the parallel efficiency, but complicates the convergence analysis. By employing some technical analysis, we provide a sharp estimate of the convergence rate, which is independent of the mesh ratio J and the distribution of the eigenvalues of the coefficient matrix. An extension of the results to problems with time-periodic conditions is also given. Several numerical experiments are carried out to verify the theoretical results.

Suggested Citation

  • Wu, Shulin, 2017. "An efficient parareal algorithm for a class of time-dependent problems with fractional Laplacian," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 329-341.
    • Handle: RePEc:eee:apmaco:v:307:y:2017:i:c:p:329-341
    DOI: 10.1016/j.amc.2017.02.012
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300317301169
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2017.02.012?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. Nicole Cusimano & Alfonso Bueno-Orovio & Ian Turner & Kevin Burrage, 2015. "On the Order of the Fractional Laplacian in Determining the Spatio-Temporal Evolution of a Space-Fractional Model of Cardiac Electrophysiology," PLOS ONE, Public Library of Science, vol. 10(12), pages 1-16, December.
    2. Rama Cont & Ekaterina Voltchkova, 2005. "A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models," Post-Print halshs-00445645, HAL.
    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. Lukas Gonon & Christoph Schwab, 2021. "Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models," Finance and Stochastics, Springer, vol. 25(4), pages 615-657, October.
    2. Song-Ping Zhu & Xin-Jiang He, 2018. "A hybrid computational approach for option pricing," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(03), pages 1-16, September.
    3. Ron Tat Lung Chan, 2016. "Adaptive Radial Basis Function Methods for Pricing Options Under Jump-Diffusion Models," Computational Economics, Springer;Society for Computational Economics, vol. 47(4), pages 623-643, April.
    4. Karel in 't Hout & Pieter Lamotte, 2022. "Efficient numerical valuation of European options under the two-asset Kou jump-diffusion model," Papers 2207.10060, arXiv.org, revised May 2023.
    5. Kyriakos Georgiou & Athanasios N. Yannacopoulos, 2023. "Probability of Default modelling with L\'evy-driven Ornstein-Uhlenbeck processes and applications in credit risk under the IFRS 9," Papers 2309.12384, arXiv.org.
    6. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    7. William A. Ramírez & Alessio Gizzi & Kevin L. Sack & Simonetta Filippi & Julius M. Guccione & Daniel E. Hurtado, 2020. "On the Role of Ionic Modeling on the Signature of Cardiac Arrhythmias for Healthy and Diseased Hearts," Mathematics, MDPI, vol. 8(12), pages 1-19, December.
    8. Moustapha Pemy, 2018. "Explicit Solutions for Optimal Resource Extraction Problems under Regime Switching L\'evy Models," Papers 1806.06105, arXiv.org.
    9. Erhan Bayraktar & Hao Xing, 2009. "Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(3), pages 505-525, December.
    10. Xu, Guoping & Zheng, Harry, 2010. "Basket options valuation for a local volatility jump-diffusion model with the asymptotic expansion method," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 415-422, December.
    11. Cl'ement M'enass'e & Peter Tankov, 2015. "Asymptotic indifference pricing in exponential L\'evy models," Papers 1502.03359, arXiv.org, revised Feb 2015.
    12. Ron Chan & Simon Hubbert, 2014. "Options pricing under the one-dimensional jump-diffusion model using the radial basis function interpolation scheme," Review of Derivatives Research, Springer, vol. 17(2), pages 161-189, July.
    13. Genin, Adrien & Tankov, Peter, 2020. "Optimal importance sampling for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 20-46.
    14. Bilel Jarraya & Abdelfettah Bouri, 2013. "A Theoretical Assessment on Optimal Asset Allocations in Insurance Industry," International Journal of Finance & Banking Studies, Center for the Strategic Studies in Business and Finance, vol. 2(4), pages 30-44, October.
    15. Shirzadi, Mohammad & Rostami, Mohammadreza & Dehghan, Mehdi & Li, Xiaolin, 2023. "American options pricing under regime-switching jump-diffusion models with meshfree finite point method," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    16. Hagspiel, Verena & Huisman, Kuno J.M. & Kort, Peter M. & Lavrutich, Maria N. & Nunes, Cláudia & Pimentel, Rita, 2020. "Technology adoption in a declining market," European Journal of Operational Research, Elsevier, vol. 285(1), pages 380-392.
    17. Reem Abdullah Aljedhi & Adem Kılıçman, 2020. "Fractional Partial Differential Equations Associated with L ê vy Stable Process," Mathematics, MDPI, vol. 8(4), pages 1-7, April.
    18. Fouladi, Somayeh & Dahaghin, Mohammad Shafi, 2022. "Numerical investigation of the variable-order fractional Sobolev equation with non-singular Mittag–Leffler kernel by finite difference and local discontinuous Galerkin methods," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    19. Yuanda Chen & Zailei Cheng & Haixu Wang, 2023. "Option Pricing for the Variance Gamma Model: A New Perspective," Papers 2306.10659, arXiv.org.
    20. Nicola Bruti-Liberati & Eckhard Platen, 2007. "Approximation of jump diffusions in finance and economics," Computational Economics, Springer;Society for Computational Economics, vol. 29(3), pages 283-312, 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:eee:apmaco:v:307:y:2017:i:c:p:329-341. 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.