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Mean-square convergence of numerical methods for random ordinary differential equations

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Abstract

The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the global and local mean-square convergence order of one-step explicit approximations is established. Then, the global mean-square convergence rates are investigated for RODE-Taylor schemes for general RODEs, Affine-RODE-Taylor schemes for RODEs with affine noise, and Itô-Taylor schemes for RODEs with Itô noise, respectively. The theoretical convergence results are demonstrated through numerical experiments.

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Acknowledgments

The authors would like to express their sincere gratitude to Prof. Dr. Arnulf Jentzen for his insightful comments. The authors would also like to thank the anonymous referees for their insightful comments, which have led to the much improved quality of this work.

Funding

Wang is partially supported by the National Natural Science Foundation of China (11101184, 11271151), Scientific and Technological Development Plan of Jilin Province (20200201251JC) and China Scholarship Council (201506175081), Cao is partially supported by the National Science Foundation (DMS1620150), Han is partially supported by Simons Foundation (Collaboration Grants for Mathematicians 429717), and Kloeden is partially supported by the National Natural Science Foundation of China (11571125).

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Authors and Affiliations

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

    Peng Wang

  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Yanzhao Cao, Xiaoying Han & Peter Kloeden

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  1. Peng Wang
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  2. Yanzhao Cao
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  3. Xiaoying Han
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  4. Peter Kloeden
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Correspondence to Xiaoying Han.

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Wang, P., Cao, Y., Han, X. et al. Mean-square convergence of numerical methods for random ordinary differential equations. Numer Algor 87, 299–333 (2021). https://doi.org/10.1007/s11075-020-00967-w

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  • DOI: https://doi.org/10.1007/s11075-020-00967-w

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