Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Uniformly Convergent Numerical Method for Singularly Perturbed Convection-Diffusion Problems

  • Received : 2019.07.17
  • Accepted : 2020.05.04
  • Published : 2020.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

A uniformly convergent numerical method is developed for solving singularly perturbed 1-D parabolic convection-diffusion problems. The developed method applies a non-standard finite difference method for the spatial derivative discretization and uses the implicit Runge-Kutta method for the semi-discrete scheme. The convergence of the method is analyzed, and it is shown to be first order convergent. To validate the applicability of the proposed method two model examples are considered and solved for different perturbation parameters and mesh sizes. The numerical and experimental results agree well with the theoretical findings.

Keywords

References

  1. J. Bear and A. Verruijt, Modeling groundwater flow and pollution, Springer Science & Business Media, 2012.
  2. F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81(3)(1973), 637-654. https://doi.org/10.1086/260062
  3. B. M. Chen-Charpentier and H. V. Kojouharov, An unconditionally positivity pre-serving scheme for advection-diffusion reaction equations, Math. Comput. Modeling, 57(9-10)(2013), 2177-2185. https://doi.org/10.1016/j.mcm.2011.05.005
  4. M. Chandru, P. Das and H. Ramos, Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data, Math. Methods Appl. Sci., 41(14)(2018), 5359-5387. https://doi.org/10.1002/mma.5067
  5. C. Clavero, J. C. Jorge and F. Lisbona, A uniformly convergent scheme on a nonuni-form mesh for convection-diffusion parabolic problems, J. Comput. Appl. Math., 154(2)(2003), 415-429. https://doi.org/10.1016/S0377-0427(02)00861-0
  6. A. Das and S. Natesan, Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh, Appl. Math. Comput., 271(2015), 168-186. https://doi.org/10.1016/j.amc.2015.08.137
  7. E. P. Doolan, J. J. M. Miller and W. H. A. Schilders, Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dublin, 1980.
  8. D. K. Edwards, J. T. Gier, K. E. Nelson and R. D. Roddick, Spectral and directional thermal radiation characteristics of selective surfaces for solar collectors, Solar Energy, 6(1)(1962), 1-8. https://doi.org/10.1016/0038-092X(62)90092-0
  9. R. E. Ewing, Problems arising in the modeling of processes for hydrocarbon recovery, The Mathematics of Reservoir Simulation, 3-34, SIAM, 1983.
  10. P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Robust computational techniques for boundary layers, Applied Mathematics (Boca Raton) 16, Chapman & Hall, 2000.
  11. A. D. Goodwin, C. F. Meares, L. H. DeRiemer, C. I. Diamanti, R. L. Goode, J. J. Baumert, D. J. Sartoris, R. L. Lantieri and H. D. Fawcett, Clinical studies with In-111 BLEDTA, a tumor-imaging conjugate of bleomycin with a bifunctional chelating agent, J. Nuclear Medicine, 22(9)(1981), 787-792.
  12. S. Gowrisankar and S. Natesan, Robust numerical scheme for singularly perturbed convection-diffusion parabolic initial-boundary-value problems on equidistributed grids, Comput. Phys. Commun., 185(2014), 2008-2019. https://doi.org/10.1016/j.cpc.2014.04.004
  13. P. F. Hahn, D. D. Stark, S. Saini, J. M. Lewis, J. Wittenberg and J. Ferrucci, Fer-rite particles for bowel contrast in MR imaging: design issues and feasibility studies, Radiology, 164(1)(1987), 37-41. https://doi.org/10.1148/radiology.164.1.3588924
  14. MK. Kadalbajoo, V. Gupta and A. Awasthi, A uniformly convergent B-spline collocation method on a nonuniform mesh for singularly perturbed one dimensional time-dependent linear convection-diffusion problem, J. Comput. Appl. Math., 220(1-2)(2008), 271-289. https://doi.org/10.1016/j.cam.2007.08.016
  15. H. Lamba and AM. Stuart, Convergence results for the MATLAB ODE23 routine, BIT, 38(4)(1998), 751-780. https://doi.org/10.1007/BF02510413
  16. B. E. Launder and D. B. Spalding, The numerical computation of turbulent flows, Numerical prediction of flow, heat transfer, turbulence and combustion, (1983), 96-116.
  17. R. E. Mickens, Nonstandard finite difference schemes, Applications of nonstandard finite difference schemes, 1-54, World Scientific, Atlanta, 1999.
  18. J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific Publishing, Singapore, 2012.
  19. M. Ndivhuwo, Numerical solution of 1-D convection-diffusion-reaction equation, The-sis, 2013.
  20. W. L. Oberkampf, T. G. Trucano and C. Hirsch, Verification, validation, and predictive capability in computational engineering and physics, Appl. Mech. Rev., 57(5)(2004), 345-384. https://doi.org/10.1115/1.1767847
  21. R. O'Malley, Singular perturbation methods for ordinary differential equations, Applied Mathematical Sciences 89, Springer-Verlag, New York, 1991.
  22. K. C. Patidar, On the use of nonstandard finite difference methods, J. Difference Equ. Appl., 11(2005), 735-758. https://doi.org/10.1080/10236190500127471
  23. S. J. Polak, C. Den Heijer, W. Schilders and P. Markowich, Semiconductor device modeling from the numerical point of view, Int. J. Numer. Meth. Engin., 24(4)(1987), 763-838. https://doi.org/10.1002/nme.1620240408
  24. J. A. Pudykiewicz, Application of adjoint tracer transport equations for evaluating source parameters, Atmospheric environment, 32(17)(1998), 3039-3050. https://doi.org/10.1016/S1352-2310(97)00480-9
  25. H. G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, convection-diffusion-reaction and flow problems, Springer-Verlag, Berlin, Heidelberg, 2008.
  26. L. F. Shampine and M. W. Reichelt, The matlab ode suite, SIAM, J. Scientific Computing, 18(1)(1997), 1-22. https://doi.org/10.1137/S1064827594276424
  27. M. M.Woldaregay and G. F. Duressa, Uniformly convergent numerical method for singularly perturbed delay parabolic differential equations arising in computational neu- roscience, Kragujevac J. Math., 46(1)(2022), 65-84.
  28. C. Yanping and L. Li-Bin, An adaptive grid method for singularly perturbed time-dependent convection-diffusion problems, Commun. Comput. Phys., 20(5)(2016), 1340-1358. https://doi.org/10.4208/cicp.240315.301215a