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Convergence of Finite Difference Schemes Applied to the Cauchy Problems of Quasi-linear Partial Differential Equations of the Normal Form

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Advances in Difference Equations and Discrete Dynamical Systems (ICDEA 2016)

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Abstract

We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of the analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a result about convergence. The main result shows convergence of consistent finite difference schemes even without stability, and therefore shows independence between stability and convergence for finite difference schemes. Our theoretical result can be realized numerically on multiple-precision arithmetic environments.

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Acknowledgements

This work is partially supported by JSPS Grant-in-Aid for scientific research (B) 25287028.

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Authors and Affiliations

  1. Center for the Promotion of Interdisciplinary Education and Research, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Nobuyuki Higashimori

  2. Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Hiroshi Fujiwara & Yuusuke Iso

Authors
  1. Nobuyuki Higashimori
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  2. Hiroshi Fujiwara
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  3. Yuusuke Iso
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Corresponding author

Correspondence to Nobuyuki Higashimori .

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Editors and Affiliations

  1. Department of Mathematics, Trinity University, San Antonio, Texas, USA

    Saber Elaydi

  2. Department of Information Science, Okayama University of Science, Okayama, Japan

    Yoshihiro Hamaya

  3. Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka, Japan

    Hideaki Matsunaga

  4. Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Klagenfurt, Austria

    Christian Pötzsche

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Higashimori, N., Fujiwara, H., Iso, Y. (2017). Convergence of Finite Difference Schemes Applied to the Cauchy Problems of Quasi-linear Partial Differential Equations of the Normal Form. In: Elaydi, S., Hamaya, Y., Matsunaga, H., Pötzsche, C. (eds) Advances in Difference Equations and Discrete Dynamical Systems. ICDEA 2016. Springer Proceedings in Mathematics & Statistics, vol 212. Springer, Singapore. https://doi.org/10.1007/978-981-10-6409-8_6

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