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

• Project/Area Number: 17K17948
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics; Computational science
• Research Institution: Saga 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: ¥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．

Research Products

• [Journal Article] Constructive error estimates for full discrete approximation of periodic solution for heat equation (2020)
  • 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
  • Peer Reviewed
• [Journal Article] Constructive error analysis of a fulldiscrete finite element method for the heat equations (2019)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Optimal Order Constructive a Priori Error Estimates for a Full Discrete Approximation of the Heat Equation (2017)
  • Author(s): Takuma Kimura, Teruya Minamoto, and Mitsuhiro T. Nakao
  • Journal Title: Reliable Computing Volume: 25 Pages: 202-212
  • 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)
• [Presentation] On the constructive error estimates of a full discrete approximation for time-periodic solution of the heat equations (2018)
  • Author(s): Takuma Kimura, Teruya Minamoto and Mitsuhiro T. Nakao
  • Organizer: The 18th International Symposium on Scientfic Computing, Computer Arithmetic, and Verified Numerical Computations
  • 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)