, Volume 90, Issue 1–2, pp 15–38 | Cite as

High order parameter-robust numerical method for time dependent singularly perturbed reaction–diffusion problems



We introduce a high order parameter-robust numerical method to solve a Dirichlet problem for one-dimensional time dependent singularly perturbed reaction-diffusion equation. A small parameter ε is multiplied with the second order spatial derivative in the equation. The parabolic boundary layers appear in the solution of the problem as the perturbation parameter ε tends to zero. To obtain the approximate solution of the problem we construct a numerical method by combining the Crank–Nicolson method on an uniform mesh in time direction, together with a hybrid scheme which is a suitable combination of a fourth order compact difference scheme and the standard central difference scheme on a generalized Shishkin mesh in spatial direction. We prove that the resulting method is parameter-robust or ε-uniform in the sense that its numerical solution converges to the exact solution uniformly well with respect to the singular perturbation parameter ε. More specifically, we prove that the numerical method is uniformly convergent of second order in time and almost fourth order in spatial variable, if the discretization parameters satisfy a non-restrictive relation. Numerical experiments are presented to validate the theoretical results and also indicate that the relation between the discretization parameters is not necessary in practice.


Singular perturbation problems Parameter-robust convergence Generalized Shishkin mesh Parabolic reaction–diffusion equations Fourth order compact scheme Crank–Nicolson method 

Mathematics Subject Classification (2000)

65M06 65M12 65M15 


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  1. 1.
    Bakhvalov NS (1969) Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. Zh Vychisl Mat Mat Fiz 9: 841–859MATHGoogle Scholar
  2. 2.
    Bujanda B, Clavero C, Gracia JL, Jorge JC (2007) A high order uniformly convergent alternating direction scheme for time dependent reaction diffusion problems. Numer Math 107: 1–25MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Clavero C, Gracia JL (2005) High order methods for elliptic and time dependent reaction diffusion singularly perturbed problems. Appl Math Comp 168(2): 1109–1127MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Clavero C, Gracia JL, Jorge JC (2005) High order numerical methods for one dimensional parabolic singulary perturbed problem with regular layers. Numer Meth PDEs 21: 149–169MATHMathSciNetGoogle Scholar
  5. 5.
    Doolan EP, Miller JJH, Schilders WHA (1980) Uniform numerical methods for problems with intial and boundary layers. Boole Press, DublinGoogle Scholar
  6. 6.
    Farrell PA, Hegarty AF, Miller JJH, O’Riordan E, Shishkin GI (2000) Robust computational techniques for boundary layers. Chapman and Hall/CRC Press, Boca RatonMATHGoogle Scholar
  7. 7.
    Hindmarsh AC, Gresho PM, Griffiths DF (1984) The stability of explicit euler time-integration for certain finite difference approximations of the multidimensional advection–diffusion equation. Int J Numer Meth Fluids 4(9): 853–897MATHCrossRefGoogle Scholar
  8. 8.
    Hemker PW, Shishkin GI, Shishkina LP (2000) ε-uniform schemes with high order time accuracy for parabolic singular perturbation problems. IMA J Numer Anal 20: 99–121MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ladyzhenskaya OA, Solonnikov VA, Ural’tseva NN (1968) Linear and quasilinear equation of parabolic type. Translations of Mathematical Monographs. American Mathematical Society, USAGoogle Scholar
  10. 10.
    Linss T, Madden N (2007) Parameter uniform approximations for time-dependent reaction–diffusion problems. Numer Meth PDEs 23(6): 1290–1300MATHMathSciNetGoogle Scholar
  11. 11.
    Linss T (2001) The necessity of Shishkin type decompositions. Appl Math Lett 14: 891–896MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Linss T, Roos HG, Vulanović R (2000) Uniform pointwise convergence on Shishkin-type meshes for quasilinear convection–diffusion problems. SIAM J Numer Anal 38: 897–912MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lunardi A (1995) Analytic semigroups and optimal regularity in parabolic problems. Progress in nonlinear differential equations and their applications, vol 16. Birkhäuser, BaselGoogle Scholar
  14. 14.
    Miller JJH, O’Riordan E, Shishkin GI, Shishkina LP (1998) Fitted mesh methods for problems with parabolic boundary layers. Math Proc R Irish Acad 98A(2): 173–190MATHMathSciNetGoogle Scholar
  15. 15.
    Miller JJH, O’Riordan E, Shishkin GI (1996) Fitted numerical methods for singular perturbation problems: Error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific Publishing, River EdgeMATHGoogle Scholar
  16. 16.
    Palencia C (1993) A stability result for sectorial operator in Banach spaces. SIAM J Numer Anal 30: 1373–1384MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Roos HG, Stynes M, Tobiska L (1996) Numerical methods for singularly perturbed differential equations. Springer, BerlinMATHGoogle Scholar
  18. 18.
    Roos HG, Linss T (1999) Sufficient conditions for uniform convergence on layer adpated grids. Computing 63: 27–45MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Shishkin GI (1989) Approximation of solutions of singularly perturbed boundary value problems with a parabolic boundary layers. USSR Comput Maths Math Phys 29(4): 1–10MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Shishkin GI (1988) A difference scheme for a singularly perturbed parabolic equation with a discontinous boundary condition. Zh Vychisl Mat i Mat Fiz 28: 1679–1692MathSciNetGoogle Scholar
  21. 21.
    Vulanović R (2001) A priori meshes for singularly perturbed quasilinear two point boundary value problems. IMA J Numer Anal 21: 349–366MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Vulanović R (2001) A high order scheme for quasilinear boundary vale problems with two small parameters. Computing 67: 287–303MATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia

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