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Rigorous numerical computations for 1D advection equations with variable coefficients

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Abstract

This paper provides a methodology of verified computing for solutions to 1D advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few results of verified numerical computations to initial-boundary value problems of hyperbolic PDEs. Our methodology is based on the spectral method and semigroup theory. The provided method in this paper is regarded as an efficient application of semigroup theory in a sequence space associated with the Fourier series of unknown functions. This is a foundational approach of verified numerical computations for hyperbolic PDEs. Numerical examples show that the rigorous error estimate showing the well-posedness of the exact solution is given with high accuracy and high speed.

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Notes

  1. The definition of \(C_0\) semigroup is given in Sect. 2.

  2. From the Hahn-Banach theorem (cf., e.g., [2]), it follows that \(F(x)\not =\emptyset \) for any \(x\in X\).

  3. The “\(*\)” denotes the discrete convolution (Cauchy product) defined by \(a*b=\left( \sum _{m\in \mathbb {Z}}a_{k-m}b_m\right) _{k\in \mathbb {Z}}\) for two bi-infinite sequences a and b.

  4. The variable coefficient c(x) is the real-valued function.

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Acknowledgements

The authors express their sincere gratitude to Dr. M. Sobajima in Tyokyo University of Science for his essential suggestion on generating the \(C_0\) semigroup on sequence spaces. The authors would also like to express their gratitude to the anonymous referees for providing valuable comments that improve the paper. This work was partially supported by JSPS Grant-in-Aid for Early-Career Scientists, No. 18K13453.

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Authors and Affiliations

  1. Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8573, Japan

    Akitoshi Takayasu & Yasunori Endo

  2. Department of Risk Engineering, University of Tsukuba, Tsukuba, Japan

    Suro Yoon

Authors
  1. Akitoshi Takayasu
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  2. Suro Yoon
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  3. Yasunori Endo
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Correspondence to Akitoshi Takayasu.

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Takayasu, A., Yoon, S. & Endo, Y. Rigorous numerical computations for 1D advection equations with variable coefficients. Japan J. Indust. Appl. Math. 36, 357–384 (2019). https://doi.org/10.1007/s13160-019-00345-7

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  • DOI: https://doi.org/10.1007/s13160-019-00345-7

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