Numerical Algorithms

, Volume 71, Issue 1, pp 139–150 | Cite as

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

  • Sunil Kumar
  • Mukesh Kumar
Original Paper


We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N −1 lnN) p ),p=1,2, on the Shishkin mesh and O(N p ),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.


Singular perturbation Coupled system Robust numerical method Layer-adapted meshes 

Mathematics Subject Classification (2010)

65L05 65L11 65L12 65L70 


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  1. 1.
    Athanasios, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)MATHGoogle Scholar
  2. 2.
    de Boor, C.: Good approximation by splines with variable knots. In: Meir, A., Sharma, A. (eds.) Spline Functions and Approximation Theory, Proceedings of the Symposium held at the University of Alberta, Edmonton, May 29–June 1, 1972. Birkhauser, Basel (1973)CrossRefGoogle Scholar
  3. 3.
    Cen, Z., Xu, A., Le, A.: A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems. J. Comput. Appl. Math. 234, 3445–3457 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chang, K.W., Howes, F.A.: Nonlinear Singular Perturbation Phenomena. Springer, New York (1984)CrossRefMATHGoogle Scholar
  5. 5.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust computational techniques for boundary layers. Chapman & Hall/CRC, Boca Raton, Florida (2000)MATHGoogle Scholar
  6. 6.
    Gajić, Z., Lim, M.T.: Optimal Control of Singularly Perturbed Linear Systems and Applications. Marcel Dekker, New York (2001)MATHGoogle Scholar
  7. 7.
    Kumar, S., Kumar, M.: Parameter-robust numerical method for a system of singularly perturbed initial value problems. Numer. Algorithms 59, 185–195 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ladde, G., Lakshmikantham, V., Vatsala, A.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985)MATHGoogle Scholar
  9. 9.
    Linss, T.: Sufficient conditions for uniform convergence on layer-adapted grids. Appl. Numer. Math. 37, 241–255 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Linss, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Vol. 95 of Lecture Notes in Mathematics. Springer, Berlin (2010)CrossRefGoogle Scholar
  11. 11.
    Meenakshi, P.M., Valarmathi, S., Miller, J.J.H.: Solving a partially singularly perturbed initial value problem on shishkin meshes. Appl. Math. Comput. 215, 3170–3180 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Rao, S.C.S., Kumar, S.: Second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems. Appl. Math. Comput. 219, 3740–3753 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Roos, H.G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations. In: Springer Series in Computational Mathematics. 2nd edn. Springer-Verlag, Berlin (2008)Google Scholar
  14. 14.
    Valarmathi, S., Miller, J.J.H.: A parameter-uniform finite difference method for singularly perturbed linear dynamical systems. Int. J. Numer. Anal. Mod. 7, 535–548 (2010)MathSciNetMATHGoogle Scholar
  15. 15.
    Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, N.J (1962)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  3. 3.Max Planck Institute for Solar System ResearchGöttingenGermany

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