본문 바로가기
KERD
논문KCI 등재

Uniformly Convergent Numerical Method for Singularly Perturbed Convection-Diffusion Problems

Gemechis File Duressa|Derartu Ayansa Turuna|Mesfin Mekuria Woldaregay
경북대학교 자연과학대학 수학과|2020.09|Kyungpook Mathematical Journal|제60권 제3호|pp. 629-645

2

조회

0

다운로드

1

인용

0

참고문헌

키워드

singular perturbationconvection-diffusion problemmethod of linenon-standard finite difference
주제분류수학

초록

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.

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.

참고문헌 (28)

  1. J. Bear, "Modeling groundwater flow and pollution" (2012), Springer Science & Business Media
  2. F. Black, "The pricing of options and corporate liabilities", J. Polit. Econ., 81(3), pp.637~654 (1973)
  3. B. M. Chen-Charpentier, "An unconditionally positivity preserving scheme for advection-diffusion reaction equations", Math. Comput. Modeling, 57(9-10), pp.2177~2185 (2013)
  4. M. Chandru, "Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data", Math. Methods Appl. Sci, 41(14), pp.5359~5387 (2018)
  5. C. Clavero, "A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems", J. Comput. Appl. Math, 154(2), pp.415~429 (2003)
  6. A. Das, "Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh", Appl. Math. Comput, 271, pp.168~186 (2015)
  7. E. P. Doolan, "Uniform numerical methods for problems with initial and boundary layers" (1980), Boole Press
  8. D. K. Edwards, "Spectral and directional thermal radiation characteristics of selective surfaces for solar collectors", Solar Energy, 6(1), pp.1~8 (1962)
  9. R. E. Ewing, "The Mathematics of Reservoir Simulation", pp.3~34 (1983), SIAM
  10. P. A. Farrell, "Robust computational techniques for boundary layers" (2000), Chapman & Hall
  11. A. D. Goodwin, "Clinical studies with In111 BLEDTA, a tumor-imaging conjugate of bleomycin with a bifunctional chelating agent", J. Nuclear Medicine, 22(9), pp.787~792 (1981)
  12. S. Gowrisankar, "Robust numerical scheme for singularly perturbed convection-diffusion parabolic initial-boundary-value problems on equidistributed grids", Comput. Phys. Commun, 185, pp.2008~2019 (2014)
  13. P. F. Hahn, "Ferrite particles for bowel contrast in MR imaging : design issues and feasibility studies", Radiology, 164(1), pp.37~41 (1987)
  14. MK. Kadalbajoo, "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), pp.271~289 (2008)
  15. H. Lamba, "Convergence results for the MATLAB ODE23 routine", BIT, 38(4), pp.751~780 (1998)
  16. B. E. Launder, "Numerical prediction of flow, heat transfer, turbulence and combustion", pp.96~116 (1983)
  17. R. E. Mickens, "Applications of nonstandard finite difference schemes", pp.1~54 (1999), World Scientific
  18. J. J. H. Miller, "Fitted numerical methods for singular perturbation problems" (2012), World Scientific Publishing
  19. M. Ndivhuwo, "Numerical solution of 1-D convection-diffusion-reaction equation", pp.2013
  20. W. L. Oberkampf, "and predictive capability in computational engineering and physics", Appl. Mech. Rev, 57(5), pp.345~384 (2004)
  21. R. O’Malley, "Singular perturbation methods for ordinary differential equations" (1991), Springer-Verlag
  22. K. C. Patidar, "On the use of nonstandard finite difference methods", J. Difference Equ. Appl., 11, pp.735~758 (2005)
  23. S. J. Polak, "Semiconductor device modeling from the numerical point of view", Int. J. Numer. Meth. Engin, 24(4), pp.763~838 (1987)
  24. J. A. Pudykiewicz, "Application of adjoint tracer transport equations for evaluating source parameters", Atmospheric environment, 32(17), pp.3039~3050 (1998)
  25. H. G. Roos, "Robust numerical methods for singularly perturbed differential equations, convection-diffusion-reaction and flow problems" (2008), Springer-Verlag
  26. L. F. Shampine, "The matlab ode suite", J. Scientific Computing, 18(1), pp.1~22 (1997), SIAM
  27. M. M. Woldaregay, "Uniformly convergent numerical method for singularly perturbed delay parabolic differential equations arising in computational neuroscience", Kragujevac J. Math, 46(1), pp.65~84
  28. C. Yanping, "An adaptive grid method for singularly perturbed timedependent convection-diffusion problems", Commun. Comput. Phys, 20(5), pp.1340~1358 (2016)

관련 논문AI추천

Higher Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Reaction-Diffusion ProblemsGemechis File Duressa, Mesfi n Mekuria Woldaregay, Worku Tilahun Anilay경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2021 · 제61권 제3호An exponentially fitted method for two parameter singularly perturbed parabolic boundary value problemsGemechis File Duressa, Tariku Birabasa Mekonnen대한수학회 · 대한수학회논문집2023 · 제38권 제1호Numerical method for a system of singularly perturbed convection diffusion equations with integral boundary conditionsAyyadurai Tamilselvan, Velusamy Raja대한수학회 · 대한수학회논문집2019 · 제34권 제3호Numerical Scheme for Two Parameter Singularly Perturbed Time-Delay Differential Difference EquationsAwoke Andargie Tiruneh, Getachew Adamu Derese, Mulunesh Amsalu Ayele경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2025 · 제65권 제2호PERFORMANCE OF RICHARDSON EXTRAPOLATION ON SOME NUMERICAL METHODS FOR A SINGULARLY PERTURBED TURNING POINT PROBLEM WHOSE SOLUTION HAS BOUNDARY LAYERSJustin B. Munyakazi, Kailash C. Patidar대한수학회 · 대한수학회지2014 · 제51권 제4호A combined method based on -spline and sinc methods for solving two-dimensional unsteady convection-diffusion problemsA. BARATI, J. Rashidinia장전수학회 · Advanced Studies in Contemporary Mathematics2016 · 제26권 제1호

이 간행물의 다른 논문 (14건)

Variants of Essential Arity for Partial FunctionsNareupanat Lekkoksung, Erkko Lehtonenpp. 423-443φ-prime Subsemimodules of Semimodules over Commutative SemiringsFatemeh Fatahi, Reza Safakishpp. 445-453Purities of Ordered Ideals of Ordered SemiringsPakorn Palakawong na Ayutthaya, Bundit Pibaljommeepp. 455-465On Diameter, Cyclomatic Number and Inverse Degree of Chemical GraphsReza Sharafdini, Ali Reza Ashrafi, Ali Ghalavandpp. 467-475On the Fekete-Szegö Problem for Starlike Functions of Complex OrderHanan Darwish, Abdel-Moniem Lashin, Bashar Al Saeedipp. 477-484
원문 보기