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Formulation of Einstein equation for stable numerical simulation

Research Project

Project/Area Number 16540126
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 InstitutionWaseda University

Principal Investigator

YONEDA Gen  Waseda University, Faculty of Science and Engineering, Associate professor (90277848)

Co-Investigator(Kenkyū-buntansha) SHINKAI Hisa-aki  Osaka Institute of Technology, Faculty of Information Science and Technology, Associate professor (30267405)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥2,350,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥150,000)
Fiscal Year 2007: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2006: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
KeywordsEinstein equation / gravitational wave / general relativity / numerical simulation / numerical relativity / 数値数学
Research Abstract

Numerical simulation of Einstein equation is essential for observation of gravitational wave and cosmology. But in real numerical simulation, especially in strong gravitational region or in very long time simulation, the violation of constraint often diverges in the time evolution. For accurate and stable simulation, reformulations of the Einstein equations are necessary. This is the formulation problem of numerical relativity. The purpose of this research is to propose theoretical framework of this criterion of this problem and to demonstrate it by a numerical experiment. As the preparations situation before the grant of this subsidy, we suggests the Constraint Amplification Factor(CAF) approach with eigenvalue analysis of constraint propagation. It may be said that this approach is succeeded to some extent, because much numerical computation result according to the theory was given. However, there remain many problems, too. A lot of situation that the numerical data diverge crawled during long time even if CAF was good. In addition, there remains the doubt whether it gives right solution if constraint preserve. In this study period, we could present many adjusted formulations with evaluation by CAF and the numerical proof for it. It quoted by many researchers of numerical relativity and produces result In addition, we succeeded in postponing calculation life more because the propagation type of the constraint avoided becoming the second. However, the calculation life was prolonged, and it diverges after long time. So further improvement will be demanded. In addition, in some case that a exact solution was known, we see that a close solution is gotten by numerical simulation. Of course it cannot be said to right evidence for calculating because this is only one example.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (19 results)

All 2008 2007 2006 2005 2004 Other

All Journal Article (12 results) (of which Peer Reviewed: 4 results) Presentation (7 results)

  • [Journal Article] Numerical Experiments of adjusted BSSN systems for controlling constraint violations2008

    • Author(s)
      Kenta Kiuchi and Hisa-aki Shinkai
    • Journal Title

      Physical Review D 77

      Pages: 44010-44010

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Numerical Experiments of adjusted BSSN systems for controlling constraint violations2008

    • Author(s)
      Kenta Kiuchi, Hisa-aki Shinkai
    • Journal Title

      Physical Review D 77

      Pages: 44010-44010

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Formulation Problem of the Einstein Equation for Numerical Simulations2006

    • Author(s)
      Gen Yoneda and Hisa-aki Shinkai
    • Journal Title

      Hyperbolic Problems:Theory, Numerics and Appli cationsis II

      Pages: 401-408

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Formulation Problem of the Einstein Equation for Numerical Simulations2006

    • Author(s)
      Gen Yoneda, Hisa-aki Shinkai
    • Journal Title

      Hyperbolic Problems : Theory, Numerics and Applicationsis. II

      Pages: 401-408

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] formulation Problem of the Einstein Equation for Numerical Simulations2006

    • Author(s)
      Gen Yoneda, Hisa-ski Shinkai
    • Journal Title

      Hyperbolic Problems : Theory, Numerics and Applicationsis.II

      Pages: 401-408

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Formulation problem in numerical relativity2005

    • Author(s)
      Hisa-aki Shinkai and Gen Yoneda
    • Journal Title

      Bulletin of the Japan Society for Industrial and Applied Mathematics 15(1)

      Pages: 1-15

    • NAID

      110001888932

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Formulation Problem in numerical relativity2005

    • Author(s)
      Hisa-aki Shinkai, Gen Yoneda
    • Journal Title

      Bulletin of the Japan Society of Industrial and Applied Mathematics 15(1)

      Pages: 1-15

    • NAID

      110001888932

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Formulation problem in numerical relativity2005

    • Author(s)
      Hisa-aki Shinkai, Gen Yoneda
    • Journal Title

      Bulletin of the Japan Society for Industrial and Applied Mathematics Vol.15 No.1(accepted for pulication)

      Pages: 2-15

    • NAID

      110001888932

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Constraint propagation in N+1 dimensional space-time2004

    • Author(s)
      Hisa-aki Shinkai and Gen Yoneda
    • Journal Title

      General Relativity and Gravitation 36(8)

      Pages: 1931-1937

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Constraint propagation in N+1 dimensional space-time2004

    • Author(s)
      Hisa-aki Shinkai, Gen Yoneda
    • Journal Title

      General Relativity and Gravitation 36(8)

      Pages: 1931-1937

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Formulation Problem of the Einstein Equation for Numerical Simulations

    • Author(s)
      Gen Yoneda, Hisa-aki Shinkai
    • Journal Title

      Hyperbolic Problems : Theory, Numerics and Applications (accepted for publication)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Formulation problem of the Einstein equation for numerical simulations

    • Author(s)
      Gen YONEDA, Hisa-aki SHINKAI
    • Journal Title

      HYP2004 Conference Proceedings (submitted)

    • Related Report
      2004 Annual Research Report
  • [Presentation] Stable numerical simulations via adjusted ADM systems2007

    • Author(s)
      Hisa-aki Shinkai and Gen Yoneda
    • Organizer
      The 2007 Spring Japan Physical Society Meeting
    • Place of Presentation
      Tokyo Metropolitan Univ.
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Controlling constraint violation using adjusted ADM systems2007

    • Author(s)
      Hisa-aki Shinkai and Gen Yoneda
    • Organizer
      The 18th International Conference on General Relativity and Gravitation
    • Place of Presentation
      Sydney
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Controlling constraint violations in numerical relativity2007

    • Author(s)
      Hisa-aki Shinkai
    • Organizer
      The 8th Asia-Pacific International Conference on Gravitation and Astrophysics(ICGA8)
    • Place of Presentation
      Nara
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Stable numerical simulations via adjusted ADM systems2007

    • Author(s)
      Hisa-aki Shinkai, Gen Yoneda
    • Organizer
      The 2007 Spring Japan Physical Society Meeting
    • Place of Presentation
      Tokyo Metropolitan Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Controlling constraint violations in numerical relativity2007

    • Author(s)
      H. Shinkai
    • Organizer
      The 8th Asia-Pacific International Conference on Gravitation and Astrophysics
    • Place of Presentation
      Nara
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Constraint propagation revisited2006

    • Author(s)
      Hisa-aki Shinkai and Gen Yoneda
    • Organizer
      The 16th Workshop on General Relativity and Gravitation
    • Place of Presentation
      Niigata
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Formulation Problem of Numerical Relativity2004

    • Author(s)
      Gen Yoneda and Hisa-aki Shinkai
    • Organizer
      Tenth International Conference on Hyperbolic Problems
    • Place of Presentation
      Osaka
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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