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International Conference on Numerical Methods and Applications

NMA 2018: Numerical Methods and Applications pp 449–457Cite as

Alternate Overlapping Schwarz Method for Singularly Perturbed Semilinear Convection-Diffusion Problems

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11189))

Abstract

We propose an alternate overlapping Schwarz method to solve the singularly perturbed semilinear convection-diffusion problems. The method decomposes the original domain into two overlapping subdomains. One, the outside boundary layer subdomain, and the other, inside boundary layer subdomain. On the outside boundary layer subdomain, a combination of the second order compact difference scheme and central difference scheme with uniform mesh is considered; while on the inside boundary layer subdomain a central difference scheme with a special piecewise-uniform mesh is considered. The convergence analysis is given and the method is shown to have almost second order parameter-uniform convergence. Numerical experiments are presented to demonstrate the efficiency of the method.

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References

  1. Allen III, M., Herrera, I., Pinder, G.: Numerical Modelling in Science and Engineering. Wiley-Interscience, New York (1988)

    Book  Google Scholar 

  2. Boglaev, I.: The solution of a singularly perturbed convection-diffusion problem by an iterative domain decomposition method. Numer. Algorithms 31, 27–46 (2002)

    Article  MathSciNet  Google Scholar 

  3. Clavero, C., Gracia, J.L., Lisbona, F.: High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type. Numer. Algorithms 22, 73–97 (1999)

    Article  MathSciNet  Google Scholar 

  4. Farrel, P.A., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: A uniformly convergent finite difference scheme for a singularly perturbed semilinear equations. SIAM J. Numer. Anal. 33(3), 1135–1149 (1996)

    Article  MathSciNet  Google Scholar 

  5. Garbey, M.: A Schwarz alternating procedure for singular perturbation problems. SIAM J. Sci. Comput. 17, 1175–1201 (1996)

    Article  MathSciNet  Google Scholar 

  6. Lorenz, J.: Stability and monotonicity properties of stiff quasilinear boundary problems. Univ. u Novom Sadu Zb. rad. Prir.-Mat. Fak. Ser. Mat. 12, 151–175 (1982)

    MathSciNet  MATH  Google Scholar 

  7. Lynch, R.E., Rice, J.R.: A high-order difference method for differential equations. Math. Comput. 34(150), 333–372 (1980)

    Article  MathSciNet  Google Scholar 

  8. O’Malley Jr., R.E.: Singular Perturbation Methods for Ordinary Differential Equations. Springer, New York (1991). https://doi.org/10.1007/978-1-4612-0977-5

    Book  MATH  Google Scholar 

  9. Mathew, T.P.: Uniform convergence of the Schwarz alternating method for solving singularly perturbed advection-diffusion equations. SIAM J. Numer. Anal. 35, 1663–1683 (1998)

    Article  MathSciNet  Google Scholar 

  10. Murray, J.D.: Lectures on Nonlinear Differential Equation Models in Biology. Clarendon Press, Oxford (1977)

    MATH  Google Scholar 

  11. Pouly, L., Pousin, J.: A spray combustion problem. Math. Models Methods Appl. Sci. 2, 237–249 (1994)

    Article  MathSciNet  Google Scholar 

  12. Rao, S.C.S., Kumar, M.: Parameter-uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction-diffusion problems. Nonlinear Anal. 71, e1579–e1588 (2009)

    Article  MathSciNet  Google Scholar 

  13. Rao, S.C.S., Kumar, S.: Robust high order convergence of a overlapping Schwarz method for singularly perturbed semilinear reaction-diffusion problem. J. Comput. Math. 31, 509–521 (2013)

    Article  MathSciNet  Google Scholar 

  14. Rao, S.C.S., Kumar, S., Kumar, M.: A parameter-uniform B-spline collocation method for singularly perturbed semilinear reaction-diffusion problems. J. Optim. Theory Appl. 146, 795–809 (2010)

    Article  MathSciNet  Google Scholar 

  15. van Roosbroeck, W.V.: Theory of flows of electrons and holes in germanium and other semiconductors. Bell Syst. Tech. J. 29, 560–607 (1950)

    Article  Google Scholar 

  16. Shishkin, G.I., Shishkina, L.P.: The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations. J. Phys. Conf. Ser. 55, 203–213 (2006)

    Article  Google Scholar 

  17. Stynes, M., Tobiska, L.: A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numer. Algorithms 18, 337–360 (1998)

    Article  MathSciNet  Google Scholar 

  18. Vulanovic, R.: A uniform numerical method for quasilinear singular perturbation problems without turning points. Computing 41, 97–106 (1989)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the valuable comments and suggestions from the anonymous referees. The research work of the second author is supported by Council of Scientific and Industrial Research, India.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110 016, India

    S. Chandra Sekhara Rao & Varsha Srivastava

Authors
  1. S. Chandra Sekhara Rao
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  2. Varsha Srivastava
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Correspondence to S. Chandra Sekhara Rao .

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Editors and Affiliations

  1. Sofia University “St. Kliment Ohridski”, Sofia, Bulgaria

    Geno Nikolov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Natalia Kolkovska

  3. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Krassimir Georgiev

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Rao, S.C.S., Srivastava, V. (2019). Alternate Overlapping Schwarz Method for Singularly Perturbed Semilinear Convection-Diffusion Problems. In: Nikolov, G., Kolkovska, N., Georgiev, K. (eds) Numerical Methods and Applications. NMA 2018. Lecture Notes in Computer Science(), vol 11189. Springer, Cham. https://doi.org/10.1007/978-3-030-10692-8_51

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  • DOI: https://doi.org/10.1007/978-3-030-10692-8_51

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10691-1

  • Online ISBN: 978-3-030-10692-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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