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Geometric numerical integrators for Hunter–Saxton-like equations

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Abstract

We present novel geometric numerical integrators for Hunter–Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter–Saxton equation, the modified Hunter–Saxton equation, and the two-component Hunter–Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators.

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Acknowledgements

We greatly appreciate the refrees’ comments on an earlier version. We would like to thank Marcus Wunsch and Xavier Raynaud for interesting discussions. DC acknowledges support from UMIT Research Lab at Umeå University and the FY2012 Researcher Exchange Program between the Japan Society for the Promotion of Science and the Royal Swedish Academy of Sciences. YM, FD and TM acknowledge supprt from JSPS KAKENHI Grant Numbers 25287030, 15H03635, 16KT0016, 16K17550, 17H02826.

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Authors and Affiliations

  1. Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan

    Yuto Miyatake

  2. Department of Mathematics and Mathematical Statistics, Umeå University, 90187, Umeå, Sweden

    David Cohen

  3. Department of Mathematics, University of Innsbruck, 6020, Innsbruck, Austria

    David Cohen

  4. Cybermedia Center, Osaka University, Machikaneyama 1-32, Toyonaka, Osaka, 560-0043, Japan

    Daisuke Furihata

  5. Department of Mathematical Informatics, Graduate School of System of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-8656, Japan

    Takayasu Matsuo

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  1. Yuto Miyatake
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  2. David Cohen
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  3. Daisuke Furihata
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  4. Takayasu Matsuo
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Correspondence to Yuto Miyatake.

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Miyatake, Y., Cohen, D., Furihata, D. et al. Geometric numerical integrators for Hunter–Saxton-like equations. Japan J. Indust. Appl. Math. 34, 441–472 (2017). https://doi.org/10.1007/s13160-017-0252-1

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  • DOI: https://doi.org/10.1007/s13160-017-0252-1

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