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

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  • 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
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    References listed on IDEAS

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    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.
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