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Precision Improvement of Numerical Simulation Including Iteration Processes for Nonlinear Evolution Equations

Research Project

Project/Area Number 14540129
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKanazawa University (2003-2004)
Kumamoto University (2002)

Principal Investigator

HATAUE Itaru  Kanazawa University, Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (50218476)

Co-Investigator(Kenkyū-buntansha) IMAI Hitoshi  Tokushima University, Department of Applied Physics and Mathematics, Professor, 工学部, 教授 (80203298)
ZHANG Shao-Liang  University of Tokyo, Department of Applied Physics, Associate professor, 工学系研究科, 助教授 (20252273)
IWASA Manabu  Kanazawa University, Graduate School of Science and Technology, Associate professor, 自然科学研究科, 助教授 (30232648)
SAISHO Yasumasa  Kanazawa University, Graduate School of Engineering, Associate professor, 工学研究科, 助教授 (70195973)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2002: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsiteration process / convergence speed / linear system / Infinite-Precision Numerical Simulation / Krylov-subspace / Multivariate Analysis / stochastic differential equation / ランダム項 / 確率差分方程式 / リアプノフ指数 / 流体シミュレーション / 移動境界問題 / 数値解の安定的構造 / 確率微分方程式
Research Abstract

For the purpose of construction of noble schemes in which influence of errors are extremely removed, we did consideration about setting of the reasonable condition for convergence of numerical solutions in iterative steps in solving the linear systems and about speed-up of convergence. In addition, we applied them to the issues of real fluid calculation and the problem of traffic jam in order to analyze several factors which govern nonlinear phenomena. First, we analyzed the structure of dynamical systems which are produced by discretizing the nonlinear differential equations from several viewpoints such as numerical experiments, visualization of nonlinear structure of attractors and statistical consideration. Concretely, we clarified essential structure of numerical solutions in discrete dynamical system and proposed the criterion to judge reliability of solutions.
Next, we tried to discuss how structure of numerical solutions of a stochastic differential equation changes by the insertion of errors with the random style from the viewpoints of probabilistic approaches. Concretely, we tried to investigate the dependence of the structure of numerical solutions on insertion of random errors. As a fundamental study, the stochastic differential equation based on the deterministic logistic differential equation was considered and the relation between the size of noise and characteristics of obtained numerical solutions was discussed. Furthermore, we tried to discuss the dependence of the structure of numerical solutions of incompressible fluid equations on insertion of random errors in solving simultaneous equations.
Next, direct numerical simulation to an integral equation of the first kind was carried out by using IPNS(Infinite-Precision Numerical Simulation). Numerical results are very satisfactory in accuracy. Moreover, they also show some interesting facts. These numerical results show IPNS facilitates numerical analysis for such inverse problems.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (21 results)

All 2005 2004 2003 2002 Other

All Journal Article (10 results) Book (1 results) Publications (10 results)

  • [Journal Article] 数値解の構造に対する誤差移入の影響についての確率論的アプローチ2005

    • Author(s)
      畑上到, 税所康正
    • Journal Title

      第54回理論応用力学講演会講演論文集

      Pages: 219-220

    • NAID

      130004603589

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Study on bilinear scheme and application to three-dimensional convective equation2004

    • Author(s)
      I.Hataue, Y.Matsuda
    • Journal Title

      Information Vol.7, No.3

      Pages: 297-310

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Study on convergency of SOR method by domain decomposition in parallel computation2004

    • Author(s)
      I.Hataue, K.Sakatani
    • Journal Title

      Information Vol.7, No.3

      Pages: 311-320

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Study on convergency of SOR method by domain decomposition parallel computation2004

    • Author(s)
      I.Hataue, K.Sakatani
    • Journal Title

      Information Vol.7, No.3

      Pages: 311-320

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the stabilizing effect of the condition for convergence in iteration processes in flow simulations2004

    • Author(s)
      I.Hataue
    • Journal Title

      Proceedings of ECCOMAS 2004

      Pages: 1-11

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On analogy and dissimilarity of dependence of stability on several parameters in flow simulations2003

    • Author(s)
      I.Hataue
    • Journal Title

      Journal of Computational and Applied Mathematics Vol.159, No.1

      Pages: 45-53

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Direct Simulation of an Integral of the First Kind2003

    • Author(s)
      H.Imai, T.Takeuchi
    • Journal Title

      Proceedings for International Conference on Inverse Problems Recent Developments in Theories and Numerics

      Pages: 247-254

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A class of product-type Krylov-subspace methods for solving nonsymmetric linear systems2002

    • Author(s)
      S.-L.Zhang
    • Journal Title

      Journal of Computational and Applied Mathematics Vol.149, No.1

      Pages: 297-305

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Concentration probabilities for restricted and unrestricted MLEs2002

    • Author(s)
      M.Iwasa, Y.Moritani
    • Journal Title

      Journal of Multivariate Analysis Vol.80, No.1

      Pages: 58-66

    • NAID

      10013877361

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A class of product-type Krylov-subspace methods for solving nonsymmetric linear systems

    • Author(s)
      S.-L.Zhang
    • Journal Title

      Journal of Computational and Applied Mathematics Vol.149, No.1

      Pages: 247-254

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] フーリエ解析とその応用2004

    • Author(s)
      畑上到
    • Total Pages
      217
    • Publisher
      数理工学社
    • Related Report
      2004 Annual Research Report
  • [Publications] I.Hataue: "On analogy and dissimilarity of dependence of stability on several parameters in flow simulations"Journal of Computational and Applied Mathematics. Vol.159,No.1. 45-53 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] I.Hataue, Y.Matsuda: "Study on bilinear scheme and application to three-dimensional convective equation"Information. Vol.7,No.3. 297-310 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] I.Hataue, K.Sakatani: "Study on convergency of SOR method by domain decomposition in parallel computation"Information. Vol.7,No.3. 311-320 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] I.Hataue, T.Nagao: "One-dimensional traffic flow by compressible flow model including several types of optimal velocity functions"Proceedings of IFAC CTS 2003 Symposium. 409-414 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] I.Hataue, O.Fujisawa: "Analysis of Influence of Share Layer on the Structure of Two-dimensional Cavity Flow in Several Reynolds Numbers"Information. Vol.6, No.1. 37-48 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] I.Hataue, T.Nagao: "A Study of One-dimensional Traffic Flow by Compressible Flow Model Including Optimal Velocity Function"Information. Vol.6, No.1. 57-70 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] I.Hataue: "On analogy and dissimilarity of dependence of stability on several parameters in flow simulations"Journal of Computational and Applied Mathematics. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Imai, T.Takeuchi: "Direct Simulation of an Integral of the First Kind"Proceedings for International Conference on Inverse Problems, Recent Developments in Theories and Numerics, World Scientific. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] S.-L.Zhang: "A class of product-type Krylov-subspace methods for solving nonsymmetric linear systems"Journal of Computational and Applied Mathematics. Vol.149,No.1. 297-305 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Iwasa, Y.Moritani: "Concentration probalities for restricted and unrestricted MLEs"Journal of Multivariate Analysis. Vol.80, No.1. 58-66 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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