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Structure-preserving numerical method for partial differential equations based on Voronoi diagram

Research Project

Project/Area Number 26610038
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionOsaka University

Principal Investigator

Furihata Daisuke  大阪大学, サイバーメディアセンター, 准教授 (80242014)

Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2015: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords構造保存数値解法 / 離散変分 / ボロノイ分割 / 非構造格子 / ボロノイ格子 / 偏微分方程式 / 局所保存則 / 大域保存則 / 離散変分導関数法
Outline of Final Research Achievements

The discretization of the differential limiting operation is the base of the original finite difference method. We appropriately introduced it to our study via the Voronoi decomposition and derived the exact discrete Green theorem by Voronoi discretization on the arbitrary lattice in the multidimensional region, e.g., the gradient, Laplacian.Our results include mathematical proofs and mathematically rigorous.
This showed that the Voronoi-Delaunay triangulation has favorable characteristics to allow arbitrary lattice placement while inheriting excellent mathematical properties.Furthermore, we adopted this Voronoi difference as the basis of the discrete variational derivative method to design numerical schemes for some partial differential equations.This indicates that our studies strengthened both the practicality and the theoretical foundation of the numerical solution method of partial differential equations.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (18 results)

All 2017 2016 2015 2014

All Journal Article (4 results) (of which Peer Reviewed: 4 results,  Acknowledgement Compliant: 1 results,  Open Access: 1 results) Presentation (14 results) (of which Int'l Joint Research: 2 results,  Invited: 4 results)

  • [Journal Article] Some Discrete Inequalities for Central-Difference Type Operators2017

    • Author(s)
      Hiroki Kojima, Takayasu Matsuo and Daisuke Furihata
    • Journal Title

      Mathematics of Computation

      Volume: 86 Issue: 306 Pages: 1719-1739

    • DOI

      10.1090/mcom/3154

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A novel discrete variational derivative method using ``average-difference methods''2016

    • Author(s)
      D. Furihata, S. Sato, T. Matsuo
    • Journal Title

      JSIAM Letters

      Volume: 8 Issue: 0 Pages: 81-84

    • DOI

      10.14495/jsiaml.8.81

    • NAID

      130005434752

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A stabilization of multistep linearly implicit schemes for dissipative systems2014

    • Author(s)
      T. Matsuo, D. Furihata
    • Journal Title

      J. Comput. Appl. Math.

      Volume: 264 Pages: 38-48

    • DOI

      10.1016/j.cam.2013.12.028

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] 離散変分導関数法 --偏微分方程式の構造保存解法--2014

    • Author(s)
      降籏 大介, 松尾 宇泰
    • Journal Title

      日本数学会「数学」

      Volume: 66 Pages: 135-156

    • NAID

      130006882546

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] A relaxation of discrete gradient2017

    • Author(s)
      Daisuke Furihata
    • Organizer
      2017 NCTS Workshop on Applied Mathematics at Tainan
    • Place of Presentation
      Tainan, Taiwan
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Relaxations of discrete gradients for differential equations2017

    • Author(s)
      Daisuke Furihata
    • Organizer
      Third International ACCA-UK/ACCA-JP Workshop
    • Place of Presentation
      London, UK
    • Related Report
      2016 Annual Research Report
  • [Presentation] Fast and asymmetric structure-preserving numerical methods based on discrete variational derivative for PDEs2016

    • Author(s)
      Daisuke Furihata
    • Organizer
      Numerical Analysis of Stochastic Partial Differential Equations
    • Place of Presentation
      Gothenburg, Sweden
    • Year and Date
      2016-09-06
    • Related Report
      2016 Annual Research Report
  • [Presentation] 陽的シングルステップ構造保存解法2016

    • Author(s)
      降籏 大介
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      北九州国際会議場
    • Related Report
      2016 Annual Research Report
  • [Presentation] Fast and structure-preserving schemes for PDEs based on discrete variational derivative method2016

    • Author(s)
      Daisuke Furihata
    • Organizer
      IMI-La Trobe Joint Conference ”Geometric Numerical Integration and its Applications”
    • Place of Presentation
      Bundoora, Australia
    • Related Report
      2016 Annual Research Report
  • [Presentation] Fast and structure-preserving numerical methods for partial differential equations2015

    • Author(s)
      Daisuke Furihata
    • Organizer
      International Conference on Scientific Computation and Differential Equations (SciCADE)
    • Place of Presentation
      Potsdam, GERMANY
    • Year and Date
      2015-09-14
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 非多項式非線形問題の構造保存数値解法とその高速化2015

    • Author(s)
      降籏 大介
    • Organizer
      日本応用数理学会2015年度年会
    • Place of Presentation
      金沢大学、金沢、日本
    • Year and Date
      2015-09-10
    • Related Report
      2015 Research-status Report
  • [Presentation] Asymmetric numerical schemes based on the discrete variational derivative method and a practical application – to treat strongly nonlinear differential equations –2015

    • Author(s)
      Daisuke Furihata
    • Organizer
      International Congress on Industrial and Applied Mathematics (ICIAM)
    • Place of Presentation
      Beiing, CHINA
    • Year and Date
      2015-08-12
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 離散変分導関数法入門 -偏微分方程式の構造保存数値解法-2015

    • Author(s)
      降籏 大介
    • Organizer
      第28回計算数理工学 forum
    • Place of Presentation
      京都大学
    • Year and Date
      2015-03-20
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Discrete variational derivative method: A structre-preserving method for partial differential equations2015

    • Author(s)
      Daisuke Furihata
    • Organizer
      The fifth workshop on computer-assisted science
    • Place of Presentation
      Osaka University
    • Year and Date
      2015-01-30
    • Related Report
      2014 Research-status Report
  • [Presentation] 離散変分導関数法入門 -意外に難しくない、微分方程式の構造数値解-2014

    • Author(s)
      降籏 大介
    • Organizer
      第6回若手による流体セミナー
    • Place of Presentation
      神戸大学
    • Year and Date
      2014-10-25
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 微分方程式に対する構造保存解法の高速化について -より良い解をより高速に求めるには-2014

    • Author(s)
      降籏 大介
    • Organizer
      RIMS研究集会 新時代の科学技術を牽引する数値解析学
    • Place of Presentation
      京都大学
    • Year and Date
      2014-10-08
    • Related Report
      2014 Research-status Report
  • [Presentation] A preditor corrector iteration method based on the discrete variational derivative method2014

    • Author(s)
      Daisuke Furihata
    • Organizer
      Kyoto Conference on Numerical Analysis and Differential Equations
    • Place of Presentation
      Kyoto University
    • Year and Date
      2014-09-19
    • Related Report
      2014 Research-status Report
  • [Presentation] Fast computation for nonlinear PDEs via a predictor-corrector iteration based on a structure preserving method2014

    • Author(s)
      Daisuke Furihata
    • Organizer
      Numerical Analysis for Partial Differential Equations
    • Place of Presentation
      東北大学
    • Year and Date
      2014-06-20
    • Related Report
      2014 Research-status Report
    • Invited

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Published: 2014-04-04   Modified: 2018-03-22  

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