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Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy functionals

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Abstract

In this paper, we present a novel investigation of the so-called SAV approach, which is a framework to construct linearly implicit geometric numerical integrators for partial differential equations with variational structure. SAV approach was originally proposed for the gradient flows that have lower-bounded nonlinear potentials such as the Allen–Cahn and Cahn–Hilliard equations, and this assumption on the energy was essential. In this paper, we propose a novel approach to address gradient flows with unbounded energy functionals such as the KdV equation by a decomposition of energy functionals. Further, we will show that the equation of the SAV approach, which is a system of equations with scalar auxiliary variables, is expressed as another gradient system that inherits the variational structure of the original system. This expression allows us to construct novel higher-order integrators by a certain class of Runge-Kutta methods. We will propose second and fourth order schemes for conservative systems in our framework and present several numerical examples.

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Acknowledgements

The first author was supported by JSPS Grant-in-Aid for Early-Career Scientists (No. 19K14590). The second author was supported by JSPS Grant-in-Aid for Research Activity Start-up (No. 19K23399).

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Authors and Affiliations

  1. Department of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8603, Japan

    Tomoya Kemmochi

  2. Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-0033, Japan

    Shun Sato

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  1. Tomoya Kemmochi
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Correspondence to Tomoya Kemmochi.

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Communicated by Christian Lubich.

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Kemmochi, T., Sato, S. Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy functionals. Bit Numer Math 62, 903–930 (2022). https://doi.org/10.1007/s10543-021-00904-w

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