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Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions

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Abstract

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.

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References

  1. Bialecki, B., Fernandes, R.I.: An orthogonal spline collocation alternating direction implicit Crank-Nicolson method for linear parabolic problems on rectangles. SIAM J. Num. Anal. 36, 1414–1434 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bialecki, B., Fernandes, R.I.: An alternating-direction implicit orthogonal spline collocation scheme for nonlinear parabolic problems on rectangular polygons. SIAM J. Sci. Comput. 28, 1054–1077 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bialecki, B., Fernandes, R.I.: Alternating direction implicit orthogonal spline collocation on non-rectangular regions. Adv. Appl. Math. Mech 5, 461–476 (2013)

    Article  MathSciNet  Google Scholar 

  4. Bialecki, B., Fernandes, R.I.: A convergence analysis of orthogonal spline collocation for solving a two-point boundary value problem without the boundary subintervals. Int. J. Numer. Anal. Mod. 13, 383–402 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Buzbee, B.L., Dorr, F.W., George, J.A., Golub, G.H.: The direct solution of the discrete Poisson equation on irregular regions. SIAM J. Numer. Anal. 8, 722–736 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diaz, J.C., Fairweather, G., Keast, P.: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination. ACM Trans. Math. Softw. 9, 358–375 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diaz, J.C., Fairweather, G., Keast, P.: Algorithm 603 COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination. ACM Trans. Math. Softw. 9, 376–380 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Düring, B., Fournié, M., Rigal, A., et al.: High-order ADI schemes for convection-diffusion equations with mixed derivative terms. In: Azaïez, M. (ed.) Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM’12, pp. 217–226. Lecture Notes in Computational Science and Engineering, p 95. Springer, Heidelberg (2013)

  9. Fairweather, G., Mitchell, A.R.: A new computational procedure for ADI methods. SIAM J. Numer. Anal. 4, 163–170 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fernandes, R.I., Bialecki, B., Fairweather, G.: Alternating direction implicit orthogonal spline collocation methods for evolution equations. In: Jacob, M. J., Panda, S (eds.) Proceedings of the International Conference on Mathematical Modelling and Applications to Industrial Problems (MMIP-2011), pp 3–11. Macmillan Publishers India Limited, Calicut (2011)

  11. Hendricks, C., Ehrhardt, M., Günther, M.: High-order-compact ADI schemes for diffusion equations with mixed derivatives in the combination technique. Appl. Numer. Math. 101, 36–52 (2016)

  12. Karaa, S.: A high-order ADI method for parabolic problems with variable coefficients. Int. J. Comput. Math. 86, 109–120 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karaa, S., Zhang, J.: High-order ADI method for solving convection-diffusion problems. J. Comput. Phys. 198, 1–9 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mitchell, A.R., Fairweather, G.: Improved forms of the alternating direction methods of Douglas, Peaceman, and Rachford for solving parabolic and elliptic equations. Numer. Math. 6, 285–292 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peaceman, D.W., Rachford Jr. H. H.: The numerical solution of elliptic and parabolic differential equations. J. Soc. Indust. Appl. Math. 3, 28–41 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  16. Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, Inc., New York (2001)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, USA

    Bernard Bialecki

  2. Department of Mathematics, The Petroleum Institute, Abu Dhabi, United Arab Emirates

    Ryan I. Fernandes

Authors
  1. Bernard Bialecki
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  2. Ryan I. Fernandes
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Correspondence to Bernard Bialecki.

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Bialecki, B., Fernandes, R.I. Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions. Numer Algor 74, 1083–1100 (2017). https://doi.org/10.1007/s11075-016-0187-7

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  • DOI: https://doi.org/10.1007/s11075-016-0187-7

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