Journal of Optimization Theory and Applications

, Volume 151, Issue 2, pp 338–352 | Cite as

Uniform Global Convergence of a Hybrid Scheme for Singularly Perturbed Reaction–Diffusion Systems



We consider a system of coupled singularly perturbed reaction–diffusion two-point boundary-value problems. A hybrid difference scheme on a piecewise-uniform Shishkin mesh is constructed for solving this system, which generates better approximations to the exact solution than the classical central difference scheme. Moreover, we prove that the method is third order uniformly convergent in the maximum norm when the singular perturbation parameter is small. Numerical experiments are conducted to validate the theoretical results.


Singular perturbation Uniformly convergent Shishkin mesh Coupled system Global solution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Thomas, G.P.: Towards an improved turbulence model for wave-current interactions. 2nd Annual Report to EU MAST-III Project the Kinematics and Dynamics of Wave–Current Interactions, Contract No MAS3-CT95-0011 (1998) Google Scholar
  2. 2.
    Kan-On, Y., Mimura, M.: Singular perturbation approach to a 3-component reaction–diffusion system arising in population dynamics. SIAM J. Math. Anal. 29, 1519–1536 (1998) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Shishkin, G.I.: Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations. Comput. Math. Math. Phys. 35, 429–446 (1995) MathSciNetMATHGoogle Scholar
  4. 4.
    Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dún Laoghaire (1980) MATHGoogle Scholar
  5. 5.
    Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008) MATHGoogle Scholar
  6. 6.
    Kadalbajoo, M.K., Patidar, K.C.: A survey of numerical techniques for solving singularly perturbed ordinary differential equations. Appl. Math. Comput. 130, 457–510 (2002) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Xenophontos, C., Oberbroeckling, L.: A numerical study on the finite element solution of singularly perturbed systems of reaction–diffusion problems. Appl. Math. Comput. 187, 351–1367 (2007) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Xenophontos, C.: The hp finite element method for singularly perturbed systems of reaction–diffusion equations. Neural Parallel Sci. Comput. 16, 337–352 (2008) MathSciNetMATHGoogle Scholar
  9. 9.
    Valanarasu, T., Ramanujam, N.: An asymptotic initial value method for boundary value problems for a system of singularly perturbed second order ordinary differential equations. Appl. Math. Comput. 147, 227–240 (2004) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Matthews, S.: Parameter robust numerical methods for a system of two coupled singularly perturbed reaction–diffusion equations. MS thesis, School of Mathematics sciences, Dublin City University (2000) Google Scholar
  11. 11.
    Matthews, S., O’Riordan, E., Shishkin, G.I.: A numerical method for a system of singularly perturbed reaction–diffusion equations. J. Comput. Appl. Math. 145, 151–166 (2002) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Rao, S.C.S., Kumar, M.: Parameter-uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction–diffusion problems. Nonlinear Anal. 71, e1579–e1588 (2009) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rao, S.C.S., Kumar, S., Kumar, M.: A parameter-uniform B-spline collocation method for singularly perturbed semilinear reaction–diffusion problems. J. Optim. Theory Appl. 146, 795–809 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Surla, K., Uzelac, Z.: A uniformly accurate spline collocation method for a normalized flux. J. Comput. Appl. Math. 166, 291–305 (2004) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Jeffries, J.S.: A singularly perturbed semilinear system. Methods Appl. Anal. 3, 157–173 (1996) MathSciNetMATHGoogle Scholar
  16. 16.
    Shishkina, L., Shishkin, G.I.: Conservative numerical method for a system of semilinear singularly perturbed parabolic reaction–diffusion equations. Math. Model. Anal. 14, 211–228 (2009) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (1984) MATHCrossRefGoogle Scholar
  18. 18.
    Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969) MATHGoogle Scholar
  19. 19.
    Bellman, R.E., Kalaba, R.E.: Quasilinearization and Nonlinear Boundary-Value Problems. Elsevier, New York (1965) MATHGoogle Scholar
  20. 20.
    Kadalbajoo, M.K., Patidar, K.C.: Spline techniques for solving singularly-perturbed nonlinear problems on nonuniform grids. J. Optim. Theory Appl. 114, 573–591 (2002) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Rao, S.C.S., Kumar, M.: B-spline collocation method for nonlinear singularly-perturbed two-point boundary-value problem. J. Optim. Theory Appl. 134, 91–105 (2007) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiHauz KhasIndia

Personalised recommendations