Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

ROBUST NUMERICAL METHOD FOR SINGULARLY PERTURBED TURNING POINT PROBLEMS WITH ROBIN TYPE BOUNDARY CONDITIONS

  • GEETHA, N. (Department of Mathematics,School of Mathematical Sciences, Bharathidasan University) ;
  • TAMILSELVAN, A. (Department of Mathematics,School of Mathematical Sciences, Bharathidasan University)
  • Received : 2017.04.11
  • Accepted : 2019.03.15
  • Published : 2019.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

We have constructed a robust numerical method on Shishkin mesh for a class of convection diffusion type turning point problems with Robin type boundary conditions. Supremum norm is used to derive error estimates which is of order O($N^{-1}$ ln N). Theoretical results are verified by providing numerical examples.

Keywords

E1MCA9_2019_v37n3_4_183_f0001.png 이미지

FIGURE 1. Numerical solution graph of Example 6.1 for various values of ε(eps) and N = 64.

E1MCA9_2019_v37n3_4_183_f0002.png 이미지

FIGURE 2. Numerical solution graph of Example 6.1 for variousvalues of ε(eps) and N = 64..

E1MCA9_2019_v37n3_4_183_f0003.png 이미지

FIGURE 3. Maximum pointwise errors as a function of N and ε for the solution U for Example 6.1

E1MCA9_2019_v37n3_4_183_f0004.png 이미지

FIGURE 4. Maximum pointwise errors as a function of N and ε for the solution U for Example 6.2

TABLE 1. Values of DN, pN for the solution component u for Example (6.1)

E1MCA9_2019_v37n3_4_183_t0001.png 이미지

TABLE 2. Values of DN, pN for the solution component u for Example (6.2)

E1MCA9_2019_v37n3_4_183_t0002.png 이미지

References

  1. L.R. Abrahamsson, A priori estimates for solutions of singular perturbations with a turning point, Studies in Applied Mathematics 56 (1977), 51-69. https://doi.org/10.1002/sapm197756151
  2. Ali.R. Ansari, Alan.F. Hegarty, Numerical solution of a convection diffusion problem with Robin boundary conditions, Journal of Computational and Applied Mathematics 156 (2003), 221-238. https://doi.org/10.1016/S0377-0427(02)00913-5
  3. V.B. Andreyev, I.A. Savin, The computation of boundary ow with uniform accuracy with respect to a small parameter, Comp. Math. Phys. 36 (1997), 1697-1692.
  4. S. Becher, H.G. Roos, Richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers, Journal of Computational and Applied Mathematics, http://dx.doi.org/10.1016/j.cam.2015.05.022.
  5. A.E. Berger, H. Han and R.B. Kellogg, A priori estimates and analysis of a numerical method for a turning point problem, Mathematics of Computation 42 (1984), 465-492. https://doi.org/10.1090/S0025-5718-1984-0736447-2
  6. E.P. Doolan, J.J.H. Miller, W.H.A. Schildres, Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dublin, 1980.
  7. P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E.O' Riordan, G.I. Shishkin, Robust computational techniques for boundary layers, Chapman and Hall/CRC Press, Boca Raton, 2000.
  8. P.A. Farrell, Sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem, SIAM J. on Numerical Analysis 25 (1988), 618-643. https://doi.org/10.1137/0725038
  9. N. Geetha, A. Tamilselvan, Parameter uniform numerical method for third order singu-larly perturbed turning point problems exhibiting boundary layers, International Journal of Applied and Computational mathematics, DOI 10.1007/s40819-015-0064-4.
  10. N. Geetha, A. Tamilselvan, Variable Mesh Spline Approximation Method for Solv-ing Second Order Singularly Perturbed Turning Point Problems with Robin Bound-ary Conditions.,International Journal of Applied and Computational Mathematics, DOI 10.1007/s40819-016-0140-4, 2016.
  11. Jia-qi Mo, Zhao-hui Wen, A class of boundary value problems for third order differential equation with a turning point, Applied Mathematics and Mechanics 31 (2010), 1027-1032. https://doi.org/10.1007/s10483-010-1338-z
  12. M.K. Kadalbajoo, K.C. Patidar, Variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer(s), Computer Mathe-matics with Applications 42 (2001), 1439-1453. https://doi.org/10.1016/S0898-1221(01)00253-X
  13. R.E.O' Malley, Introduction to singular perturbations, Academic Press, New York, 1974.
  14. J. Mohapatra, S. Natesan, Parameter uniform numerical methods for singularly perturbed mixed boundary value problems using grid equidistribution, J. Appl. Math. Comput. 37 (2011), 247-265. https://doi.org/10.1007/s12190-010-0432-5
  15. J.J.H. Miller, E.O'Riordan, G.I. Shishkin, Fitted numerical methods for singularly per-turbed problems. Error Estimates in the minimum norm for linear problems in one and two dimensions, Revised Edition, world scientific publishing Co. Pvt. Ltd., Singapore, 2012.
  16. R. Mythili Priyadharshini, N. Ramanujam, Approximation of derivative for a singularly perturbed second-order ordinary differential equation of Robin type with discontinuous con-vection coefficient and source term, Numer. Math. Theor. Meth. Appl. 2 (2009), 100-118.
  17. S. Natesan, N. Ramanujam, A computational method for solving singularly perturbed turn-ing point problems exhibiting twin boundary layers, Applied Mathematics and Computation 93 (1998), 259-275. https://doi.org/10.1016/S0096-3003(97)10056-X
  18. S. Natesan, N. Ramanujam, Initial-Value Technique for Singularly Perturbed Turning Point Problems Exhibiting Twin Boundary Layers, Journal of Optimization Theory and Applications 99 (1998), 37-52. https://doi.org/10.1023/A:1021744025980
  19. S. Natesan, J. Jayakumar, J. Vigo-Aguiar, Parameter uniform numerical method for sin-gularly perturbed turning point problems exhibiting boundary layers, Journal Computational and Applied Mathematics 158 (2003), 121-134. https://doi.org/10.1016/S0377-0427(03)00476-X
  20. Pratibhamoy Das, Srinivasan Natesan, Higher order parameter uniform convergent schemes for robin type reaction-diffusion problems using adaptively generated grid, International Journal of Computational Methods 9 (2012), 125-152.
  21. E. O'Riordan, J. Quinn, A Singularly perturbed convection diffusion turning point problem with an interior layer, Comput. Methods. Appl. Math. 12 (2012), 206-220. https://doi.org/10.2478/cmam-2012-0012
  22. H.G. Roos, M. Stynes, L. Tobiska, Numerical methods for singularly perturbed differential equations-convection-diffusion and flow problems, Springer, 2006.
  23. Kapil K. Sharma, Pratima Rai, Kailash C. Patidar, A review on singularly perturbed differential equations with turning points and interiror layers, Applied Mathematics and Computation 219 (2013), 10575-10609. https://doi.org/10.1016/j.amc.2013.04.049
  24. W. Wasow, Linear turning point theory, Springer Verlag, New York, 1984.
  25. A.M. Watts, A Singular perturbation problem with a turning point, Bull. Austral. Math. Soc. 5 (1971), 61-73. https://doi.org/10.1017/S0004972700046888