Numerical Algorithms

, Volume 59, Issue 2, pp 185–195 | Cite as

Parameter-robust numerical method for a system of singularly perturbed initial value problems

Original Paper

Abstract

In this work we study a system of M( ≥ 2) first-order singularly perturbed ordinary differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. A maximum principle does not, in general, hold for this system. It is discretized using backward Euler difference scheme for which a general convergence result is derived that allows to establish nodal convergence of O(N − 1ln N) on the Shishkin mesh and O(N − 1) on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical experiments are performed to support the theoretical results.

Keywords

Singular perturbation Initial-value problem Layer-resolving meshes Parameter-robust convergence 

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References

  1. 1.
    Athanasios, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)MATHGoogle Scholar
  2. 2.
    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 (2000)MATHGoogle Scholar
  3. 3.
    Gajić, Z., Lim, M.-T.: Optimal Control of Singularly Perturbed Linear Systems and Applications. Marcel Dekker, New York (2001)MATHGoogle Scholar
  4. 4.
    Hemavathi, S., Bhuvaneswari, T., Valarmathi, S., Miller, J.: A parameter uniform numerical method for a system of singularly perturbed ordinary differential equations. Appl. Math. Comput. 191, 1–11 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)MATHGoogle Scholar
  6. 6.
    Linss, T.: Layer-adapted meshes for reaction-convection-diffusion problems. In: Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010)Google Scholar
  7. 7.
    Linss, T., Madden, N.: Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems. IMA J. Numer. Anal. 29, 109–125 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Meenakshi, P.M., Valarmathi, S., Miller, J.: Solving a partially singularly perturbed initial value problem on shishkin meshes. Appl. Math. Comput. 215, 3170–3180 (2010)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs (1967)Google Scholar
  10. 10.
    Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations. In: Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2008)Google Scholar
  11. 11.
    Valarmathi, S., Miller, J.: A parameter-uniform finite difference method for singularly perturbed linear dynamical systems. Int. J. Numer. Anal. Model. 7, 535–548 (2010)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia
  2. 2.Institute of Applied Mathematics and Information Technology-CNRPaviaItaly

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