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Numerical verification method of solutions for evolutionary equations by applying the matrix exponential

Research Project

Project/Area Number 17K17948
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Computational science
Research InstitutionSaga University

Principal Investigator

Kimura Takuma  佐賀大学, 理工学部, 准教授 (60581618)

Project Period (FY) 2017-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords精度保証付き数値計算法 / 微分方程式 / 有限要素法 / 発展方程式 / 数値計算 / 数値解析 / 精度保証付き数値計算
Outline of Final Research Achievements


In this research, we studied some numerical verification methods for a full discrete solution of the evolutionary partial differential equations. Proposed methods are based on the error estimates of finite element semidiscretization in space direction combined with an interpolation in time that uses the fundamental solution (represented by the exponential of matrix) for the semidiscretized problem.

Academic Significance and Societal Importance of the Research Achievements

計算機を用いた数値計算は非常に有用であるが,一般に,そこで得られる数値解は誤差を含む.そのため,数値解の精度を厳密に保証できる数値計算手法が近年活発に研究されている.
本研究の主な成果は,いくつかの条件が整えば発展方程式の数値解に対する誤差評価を与えるというものであり,与えられた問題の解の存在範囲もしくは一意存在の範囲を計算機を用いて自動的に求める精度保証付き数値計算法の開発に寄与する成果といえる.

Report

(5 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (6 results)

All 2020 2019 2018 2017

All Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Open Access: 1 results) Presentation (3 results) (of which Int'l Joint Research: 1 results)

  • [Journal Article] Constructive error estimates for full discrete approximation of periodic solution for heat equation2020

    • Author(s)
      Kimura Takuma、Minamoto Teruya、Nakao Mitsuhiro T.
    • Journal Title

      Journal of Computational and Applied Mathematics

      Volume: 368 Pages: 112510-112510

    • DOI

      10.1016/j.cam.2019.112510

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Constructive error analysis of a fulldiscrete finite element method for the heat equations2019

    • Author(s)
      Hashimoto, K., Kimura, T., Minamoto, T., Nakao, M.T.
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 36 Issue: 3 Pages: 777-790

    • DOI

      10.1007/s13160-019-00362-6

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Optimal Order Constructive a Priori Error Estimates for a Full Discrete Approximation of the Heat Equation2017

    • Author(s)
      Takuma Kimura, Teruya Minamoto, and Mitsuhiro T. Nakao
    • Journal Title

      Reliable Computing

      Volume: 25 Pages: 202-212

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] 線形熱方程式の時間周期解に対する近似解の事前誤差評価について2019

    • Author(s)
      木村 拓馬, 皆本 晃弥,中尾 充宏
    • Organizer
      第3回 精度保証付き数値計算の実問題への応用研究集会 (NVR 2019)
    • Related Report
      2019 Research-status Report
  • [Presentation] On the constructive error estimates of a full discrete approximation for time-periodic solution of the heat equations2018

    • Author(s)
      Takuma Kimura, Teruya Minamoto and Mitsuhiro T. Nakao
    • Organizer
      The 18th International Symposium on Scientfic Computing, Computer Arithmetic, and Verified Numerical Computations
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] 線形熱方程式の時間周期解に対する近似解の誤差評価について2018

    • Author(s)
      木村 拓馬, 皆本 晃弥, 中尾 充宏
    • Organizer
      第2回 精度保証付き数値計算の実問題への応用研究集会 (NVR 2018)
    • Related Report
      2018 Research-status Report

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Published: 2017-04-28   Modified: 2022-01-27  

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