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Fourth order algorithms for a semilinear singular perturbation problem

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Abstract

Fourth order finite-difference algorithms for a semilinear singularly perturbed reaction–diffusion problem are discussed and compared both theoretically and numerically. One of them is the method of Sun and Stynes (1995) which uses a piecewise equidistant discretization mesh of Shishkin type. Another one is a simplified version of Vulanović's method (1993), based on a discretization mesh of Bakhvalov type. It is shown that the Bakhvalov mesh produces much better numerical results.

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References

  1. O. Axelsson and L. Kolotilina, Monotonicity and discretization error estimates, SIAM J. Numer. Anal. 27 (1990) 1591–1611.

    Article  MATH  MathSciNet  Google Scholar 

  2. N.S. Bakhvalov, Towards optimization of methods for solving boundary value problems in presence of a boundary layer, Zh. Vychisl. Mat. i Mat. Fiz. 9 (1969) 841–859 (in Russian).

    MATH  Google Scholar 

  3. E. Bohl, Finite Modelle Gewöhnlicher Randwertaufgaben (Teubner, Stuttgart, 1981).

  4. K.W. Chang and F.A. Howes, Nonlinear Singular Perturbation Phenomena: Theory and Application (Springer, New York, 1984).

  5. R.C.Y. Chin and R. Krasny, A hybrid asymptotic-finite element method for stiff two-point boundary value problems, SIAM J. Sci. Statist. Comput. 4 (1983) 229–243.

    Article  MATH  MathSciNet  Google Scholar 

  6. P.G. Ciarlet, Discrete maximum principle for finite-difference operators, Aequationes Math. 4 (1970) 338–352.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Herceg, Uniform fourth order difference scheme for a singular perturbation problem, Numer. Math. 56 (1990) 675–693.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Herceg, R. Vulanović and N. Petrović, Higher order schemes and Richardson extrapolation for singular perturbation problems, Bull. Austral. Math. Soc. 39 (1989) 129–139.

    Article  MATH  MathSciNet  Google Scholar 

  9. J.J.H. Miller, E. O'Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems (World Scientific, Singapore, 1996).

  10. H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer, Berlin, 1996).

  11. G.I. Shishkin, Grid approximation of singularly perturbed parabolic equations with internal layers, Sov. J. Numer. Anal. Math. Modelling 3 (1988) 392–407.

    MathSciNet  Google Scholar 

  12. M. Stynes, Private communication (1996-1997).

  13. G. Sun and M. Stynes, An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem, Numer. Math. 70 (1995) 487–500.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Sun and M. Stynes, A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Math. Comp. 65 (1996) 1085–1109.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Varga, On diagonal dominance arguments for bounding ||A-1||∞, Linear Algebra Appl. 14 (1976) 211–217.

    Article  MATH  Google Scholar 

  16. R. Vulanović, On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh, Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 13 (1983) 187–201.

    MathSciNet  MATH  Google Scholar 

  17. R. Vulanović, Non-equidistant finite-difference methods for elliptic singular perturbation problems, in: Computational Methods for Boundary and Interior Layers in Several Dimensions, ed. J.J.H. Miller (Boole Press, Dublin, 1991) pp. 203–223.

  18. R. Vulanović, On numerical solution of semilinear singular perturbation problems by using the Hermite scheme, Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 23(2) (1993) 363–379.

    MATH  Google Scholar 

  19. R. Vulanović, A uniform convergence result for semilinear singular perturbation problems violating the standard stability condition, Appl. Numer. Math. 16 (1995) 383–399.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Vulanović and D. Herceg, The Hermite scheme for semilinear singular perturbation problems, J. Comput. Math. 11 (1993) 162–171.

    MathSciNet  MATH  Google Scholar 

  21. R. Vulanović, D. Herceg and N. Petrović, On the extrapolation for a singularly perturbed boundary value problem, Computing 36 (1986) 69–79.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Kent State University, Stark Campus, Canton, OH, 44720-7599, USA

    Relja Vulanović

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  1. Relja Vulanović
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Vulanović, R. Fourth order algorithms for a semilinear singular perturbation problem. Numerical Algorithms 16, 117–128 (1997). https://doi.org/10.1023/A:1019187013584

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