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CAS Application to the Construction of High-Order Difference Schemes for Solving Poisson Equation

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Computer Algebra in Scientific Computing (CASC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8660))

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Abstract

In the present work, a computer algebra system (CAS) is applied for constructing a new difference scheme of high-accuracy order for solving boundary-value problem for Poisson equation. The formulae of the difference scheme are constructed in the symbol form in CAS. CAS are used for translation of complex formulae to C++ language operators, calculation of arithmetic values of the constructed scheme coefficients and matrix elements of a system of linear algebraic equations for the discrete problem approximating the initial difference problem. Efficiency of the CAS application and of the schemes constructed with its help is shown.

The work was supported by the Russian Foundation for Basic Research (grant No. 13-01-00277).

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References

  1. Albensoeder, S., Kuhlmann, H.C.: Accurate three-dimensional lid-driven cavity flow. J. Comput. Phys. 206(2), 536–558 (2005)

    Article  MATH  Google Scholar 

  2. Botella, O., Peyret, R.: Benchmark spectral results on the lid-driven cavity flow. Comput. Fluids 27(4), 421–433 (1998)

    Article  MATH  Google Scholar 

  3. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer (1988)

    Google Scholar 

  4. Ganzha, V.G., Mazurik, S.I., Shapeev, V.P.: Symbolic manipulations on a computer and their application to generation and investigation of difference schemes. In: Caviness, B.F. (ed.) ISSAC 1985 and EUROCAL 1985. LNCS, vol. 204, pp. 335–347. Springer, Heidelberg (1985)

    Google Scholar 

  5. Gerdt, V.P., Blinkov, Y.A., Mozzhilkin, V.V.: Grobner Bases and Generation of Difference Schemes for Partial Differential Equations. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2, 051 (2006) arXiv:math.RA/0605334

    Google Scholar 

  6. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  7. Isaev, V.I., Shapeev, V.P.: High-Accuracy Versions of the Collocations and Least Squares Method for the Numerical Solution of the Navier-Stokes Equations. Comput. Math. and Math. Phys. 50(10), 1670–1681 (2010)

    Article  MathSciNet  Google Scholar 

  8. Lee, M.K., Malaya, N., Moser, R.D.: Petascale direct numerical simulation of turbulent channel flow on up to 786K cores. In: SC 2013 Proceedings of SC13: International Conference for High Performance Computing, Networking, Storage and Analysis, Denver, CO (2013)

    Google Scholar 

  9. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 102(1), 16–42 (1992)

    Article  MathSciNet  Google Scholar 

  10. Lipavskii, M.V., Tolstykh, A.I.: Tenth-order accurate multioperator scheme and its application in direct numerical simulation. Comput. Math. and Math. Phys. 53(4), 455–468 (2013)

    Article  MathSciNet  Google Scholar 

  11. Shapeev, A.V., Lin, P.: An asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows. SIAM J. Sci. Comput. 31(3), 1874–1900 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Shapeev, A.V., Shapeev, V.P.: Difference schemes of increased order of accuracy for solving elliptical equations in domain with curvilinear boundary. Comput. Math. and Math. Phys. 40(2), 213–221 (2000)

    MATH  MathSciNet  Google Scholar 

  13. Shapeev, V.P., Vorozhtsov, E.V.: CAS application to the construction of the collocations and least residuals method for the solution of 3D Navier-Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 381–392. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  14. Steinberg, S.: A problem solving environment for numerical partial differential equations. In: 6th IMACS Int. Conf. on Applications of Computer Algebra. Abstracts, St.Petersburg, Russia, June 25-28, pp. 98–99 (2000)

    Google Scholar 

  15. Stetter, H.J.: Condition analysis of overdetermined algebraic problems. Computer Algebra in Scientific Computing, pp. 345–365. Springer, Berlin (2000)

    Google Scholar 

  16. Valiullin, A.N., Shapeev, V.P., et al.: Symbolic manipulations in the methods of mathematical physics. In: Symposium Mathematics for Computer Science, Paris, March 16-18, pp. 431–438 (1982)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Novosibirsk State University, Novosibirsk, Russia

    Grigoriy M. Drozdov & Vasily P. Shapeev

  2. Khristianovich Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Vasily P. Shapeev

Authors
  1. Grigoriy M. Drozdov
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  2. Vasily P. Shapeev
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Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies (LIT), Joint Institute for Nuclear Research, 141980, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, 34132, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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© 2014 Springer International Publishing Switzerland

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Drozdov, G.M., Shapeev, V.P. (2014). CAS Application to the Construction of High-Order Difference Schemes for Solving Poisson Equation. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_8

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  • DOI: https://doi.org/10.1007/978-3-319-10515-4_8

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10514-7

  • Online ISBN: 978-3-319-10515-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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