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A-stable Runge–Kutta methods for stiff stochastic differential equations with multiplicative noise

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Abstract

In this paper, a class of one-stage semi-implicit stochastic Runge–Kutta (SISRK) methods is proposed for stiff systems with multiplicative noise. The coefficient families of SISRK methods of strong order one-half are calculated. The stability functions of these methods, applied to a scalar linear test equation with multi-dimensional multiplicative noise, are determined and their regions of stability are then compared with the corresponding stability regions of the test equation. Furthermore, we also investigate mean square stability (MS-stability) of these methods applied to two linear multi-dimensional stochastic differential test equations with multi-dimensional multiplicative noise. In particular, some SISRK methods with mean-square A-stability are derived for systems with multiplicative noise. The stability properties and numerical examples are given to illustrate the efficiency and performance of the proposed methods.

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Acknowledgments

The author is grateful to Yong Li, Jialin Hong and Jungong Xue for helpful discussions. He also thanks anonymous referees for careful reading and many helpful suggestions to improve the presentation of this paper.

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Authors and Affiliations

  1. Institute of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Peng Wang

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  1. Peng Wang
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Correspondence to Peng Wang.

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Communicated by Josselin Garnier.

This work is partially supported by NSFC (Nos. 11101184, 11271151), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20090061120038), the Science Foundation for Young Scientists of Jilin Province (No. 20130522101JH) and State Key Laboratory of Scientific and Engineering Computing, AMSS, CAS.

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Wang, P. A-stable Runge–Kutta methods for stiff stochastic differential equations with multiplicative noise. Comp. Appl. Math. 34, 773–792 (2015). https://doi.org/10.1007/s40314-014-0140-0

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  • DOI: https://doi.org/10.1007/s40314-014-0140-0

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