IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i8p1243-d391976.html
   My bibliography  Save this article

Iterative and Noniterative Splitting Methods of the Stochastic Burgers’ Equation: Theory and Application

Author

Listed:
  • Jürgen Geiser

    (The Institute of Theoretical Electrical Engineering, Ruhr University of Bochum, Universitätsstrasse 150, D-44801 Bochum, Germany)

Abstract

In this paper, we discuss iterative and noniterative splitting methods, in theory and application, to solve stochastic Burgers’ equations in an inviscid form. We present the noniterative splitting methods, which are given as Lie–Trotter and Strang-splitting methods, and we then extend them to deterministic–stochastic splitting approaches. We also discuss the iterative splitting methods, which are based on Picard’s iterative schemes in deterministic–stochastic versions. The numerical approaches are discussed with respect to decomping deterministic and stochastic behaviours, and we describe the underlying numerical analysis. We present numerical experiments based on the nonlinearity of Burgers’ equation, and we show the benefits of the iterative splitting approaches as efficient and accurate solver methods.

Suggested Citation

  • Jürgen Geiser, 2020. "Iterative and Noniterative Splitting Methods of the Stochastic Burgers’ Equation: Theory and Application," Mathematics, MDPI, vol. 8(8), pages 1-28, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1243-:d:391976
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/8/1243/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/8/1243/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Julien Baptiste & Julien Grepat & Emmanuel Lepinette, 2018. "Approximation of Non-Lipschitz SDEs by Picard Iterations," Applied Mathematical Finance, Taylor & Francis Journals, vol. 25(2), pages 148-179, March.
    2. Syoiti Ninomiya & Nicolas Victoir, 2008. "Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(2), pages 107-121.
    3. Jürgen Geiser, 2011. "Computing Exponential for Iterative Splitting Methods: Algorithms and Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-27, March.
    4. Chun-Ku Kuo & Sen-Yung Lee, 2015. "A New Exact Solution of Burgers’ Equation with Linearized Solution," Mathematical Problems in Engineering, Hindawi, vol. 2015, pages 1-7, August.
    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 Bayer & Peter K. Friz & Paul Gassiat & Jorg Martin & Benjamin Stemper, 2020. "A regularity structure for rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 782-832, July.
    2. Shigeto Kusuoka & Mariko Ninomiya & Syoiti Ninomiya, 2012. "Application Of The Kusuoka Approximation To Barrier Options," CARF F-Series CARF-F-277, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Jean-Franc{c}ois Chassagneux & Junchao Chen & Noufel Frikha, 2022. "Deep Runge-Kutta schemes for BSDEs," Papers 2212.14372, arXiv.org.
    4. Mariko Ninomiya & Syoiti Ninomiya, 2009. "A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method," Finance and Stochastics, Springer, vol. 13(3), pages 415-443, September.
    5. Masahiro Nishiba, 2013. "Pricing Exotic Options and American Options: A Multidimensional Asymptotic Expansion Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 20(2), pages 147-182, May.
    6. Kumari, Pinki & Gupta, R.K. & Kumar, Sachin, 2021. "Non-auto-Bäcklund transformation and novel abundant explicit exact solutions of the variable coefficients Burger equation," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    7. Kenichiro Shiraya & Akihiko Takahashi & Masashi Toda, 2010. "Pricing Barrier and Average Options under Stochastic Volatility Environment," CIRJE F-Series CIRJE-F-745, CIRJE, Faculty of Economics, University of Tokyo.
    8. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.
    9. Goodell, John W. & Kumar, Satish & Lim, Weng Marc & Pattnaik, Debidutta, 2021. "Artificial intelligence and machine learning in finance: Identifying foundations, themes, and research clusters from bibliometric analysis," Journal of Behavioral and Experimental Finance, Elsevier, vol. 32(C).
    10. Al Gerbi Anis & Jourdain Benjamin & Clément Emmanuelle, 2016. "Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators," Monte Carlo Methods and Applications, De Gruyter, vol. 22(3), pages 197-228, September.
    11. Abdelkoddousse Ahdida & Aur'elien Alfonsi, 2010. "Exact and high order discretization schemes for Wishart processes and their affine extensions," Papers 1006.2281, arXiv.org, revised Mar 2013.
    12. Rey, Clément, 2019. "Approximation of Markov semigroups in total variation distance under an irregular setting: An application to the CIR process," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 539-571.
    13. Alfonsi, Aurélien, 2025. "Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation," Stochastic Processes and their Applications, Elsevier, vol. 181(C).
    14. Abdelkoddousse Ahdida & Aurélien Alfonsi, 2013. "Exact and high order discretization schemes for Wishart processes and their affine extensions," Post-Print hal-00491371, HAL.
    15. Ahdida, Abdelkoddousse & Alfonsi, Aurélien, 2013. "A mean-reverting SDE on correlation matrices," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1472-1520.
    16. Kazuhiro Yoshikawa, 2015. "An Approximation Scheme for Diffusion Processes Based on an Antisymmetric Calculus over Wiener Space," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(2), pages 185-207, May.
    17. Anis Al Gerbi & Benjamin Jourdain & Emmanuelle Cl'ement, 2015. "Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators," Papers 1508.06492, arXiv.org, revised Oct 2015.
    18. Christian Bayer & Peter K. Friz, 2013. "Cubature on Wiener space: pathwise convergence," Papers 1304.4623, arXiv.org.
    19. Akiyama, Naho & Yamada, Toshihiro, 2024. "A weak approximation for Bismut’s formula: An algorithmic differentiation method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 386-396.
    20. Abdelkoddousse Ahdida & Aur'elien Alfonsi, 2011. "A Mean-Reverting SDE on Correlation matrices," Papers 1108.5264, arXiv.org, revised Feb 2012.

    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:8:y:2020:i:8:p:1243-:d:391976. 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.