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On the long-time stability of a temporal discretization scheme for the three dimensional viscous primitive equations

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Abstract

In this article, a semi-discretized Euler scheme to solve the three dimensional viscous primitive equations is studied. Based on suitable assumptions on the initial data and forcing terms, the long-time stability of the proposed scheme is proven by showing that the \(H^1\) norm (in space variables) of the solutions is bounded at each time step when the time step satisfies certain smallness condition.

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Acknowledgements

Chun-Hsiung Hsia and Ming-Cheng Shiue were partially supported by the Ministry of Science and Technology, Taiwan under grant MOST 104-2628-M-002-007-MY3 and MOST 104-2115-M-009-012-MY2 (MOST 106-2115-M-009 -011 -MY2) respectively. The authors would like to thank Professor Jie Shen for his very useful feedbacks when Hsia delivered a talk in the first draft of this research project. The authors also appreciate Professor Roger Temam for his comments and kind supports during Shiue’s visit to ISCAM at Indiana University.

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Authors and Affiliations

  1. Department of Mathematics, Institute of Applied Mathematical Sciences, National Center for Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd, Taipei, 10617, Taiwan

    Chun-Hsiung Hsia

  2. Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu, 300, Taiwan

    Ming-Cheng Shiue

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  1. Chun-Hsiung Hsia
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Correspondence to Ming-Cheng Shiue.

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Hsia, CH., Shiue, MC. On the long-time stability of a temporal discretization scheme for the three dimensional viscous primitive equations. Numer. Math. 139, 187–245 (2018). https://doi.org/10.1007/s00211-017-0934-2

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  • DOI: https://doi.org/10.1007/s00211-017-0934-2

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