Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type

Original Paper


In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques.


Parabolic 2D coupled systems Reaction-diffusion Additive schemes Shishkin meshes Uniform convergence Multigrid methods 

Mathematics Subject Classification (2010)

65M065 65N06 65N12 


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The authors thank the referees for their valuable suggestions which have helped to improve the presentation of this paper.


  1. 1.
    Clavero, C., Gracia, J.L.: Uniformly convergent additive finite difference schemes for singularly perturbed parabolic reaction-diffusion system. Comput. Math. Appl. 67, 655–670 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Clavero, C., Gracia, J.L., Jorge, J.C.: High–order numerical methods for one–dimensional parabolic singularly perturbed problems with regular layers. Numer. Methods Partial Diff. Equ. 21, 149–169 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Clavero, C., Gracia, J.L., Lisbona, F.: Second order uniform approximations for the solution of time dependent singularly perturbed reaction-diffusion systems. Int. J. Numer. Anal. Mod. 7, 428–443 (2010)MathSciNetMATHGoogle Scholar
  4. 4.
    Clavero, C., Gracia, J.L., O’Riordan, E.: A parameter robust numerical method for a two dimensional reaction-diffusion problem. Math. Comp. 74, 1743–1758 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Clavero, C., Jorge, J.C., Lisbona, F., Shishkin, G.I.: An alternating direction scheme on a nonuiform mesh for reaction-diffuion parabolic problems. IMA J. Numer. Anal. 20(2), 263–280 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers, Applied Mathematics, pp. 16. Chapman and Hall/CRC (2000)Google Scholar
  7. 7.
    Gaspar, F.J., Clavero, C., Lisbona, F.: Multigrid Methods and Finite Difference Schemes for 2D Singularly Perturbed Problems. Numerical Analysis and its Applications (Rousse, 2000), pp. 316–324, Lecture Notes in Comput. Sci. Springer, Berlin (1988). 2001Google Scholar
  8. 8.
    Gaspar, F.J., Clavero, C., Lisbona, F.: Some numerical experiments with multigrid methods on Shishkin meshes. J. Comput. Appl. Math. 138, 21–35 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gracia, J.L., Lisbona, F.: A uniformly convergent scheme for a system of reaction–diffusion equations. J. Comp. Appl. Math. 206, 1–16 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gracia, J.L., Lisbona, F., O’Riordan, E.: A coupled system of singularly perturbed parabolic reaction-diffusion equations. Adv. Comput. Math. 32, 43–61 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gracia, J.L., Madden, N., Nhan, T.A.: Applying a patched mesh method to efficiently solve a singularly perturbed reaction-diffusion problem. To appear in Modeling, Simulation and Optimization of Complex Processes, Springer (proceedings of the 6th International Conference on High Performance Scientific Computing Vietnan (2015)Google Scholar
  12. 12.
    Kellogg, R.B., Linss, T., Stynes, M.: A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions. Math. Comput. 774, 2085–2096 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kellogg, R.B., Madden, N., Stynes, M.: A parameter robust numerical method for a system of reaction-diffusion equations in two dimensions. Numer. Methods Partial Diff. Equ. 24, 312–334 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Linss, T., Stynes, M.: Numerical solution of systems of singularly perturbed differential equations. Comput. Methods Appl. Math. 9, 165–191 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    MacLachlan, S., Madden, N.: Robust solution of singularly perturbed problems using multigrid methods. SIAM J. Sci. Comput. 35, A2225–A2254 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Madden, N., Stynes, M.: A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems. IMA J. Numer. Anal. 23, 627–644 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Protter, M.H., Weinberger, H.F.: Maximum Principle in Differential Equations. Prentice-Hall, Englewood Cliffs (1967)MATHGoogle Scholar
  19. 19.
    Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, Berlin (2008)MATHGoogle Scholar
  20. 20.
    Shishkin, G.I.: Approximation of systems of singularly perurbed elliptic reaction-diffusion equations with two parameters. Comput. Math. Math. Phys. 47, 797–828 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Shishkin, G.I., Shishkina, L.P.: Approximation of a system of singularly perturbed parabolic reaction-diffusion equations in a rectangle. Zh. Vychisl. Mat. Mat. Fiz. 48, 660–673 (2008). (in Russian); translation in Comput. Math. Math. Phys. 48: 627–640 (2008)MathSciNetMATHGoogle Scholar
  22. 22.
    Shishkina, L.P., Shishkin, G.I.: Robust numerical method for a system of singularly perturbed parabolic reaction-diffusion equations on a rectangle. Math. Model. Anal. 13, 251–261 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, Inc., San Diego (2001)MATHGoogle Scholar
  24. 24.
    Vabishchevich, P.N.: Additive operator-difference schemes. Splitting schemes. De Gruyter, Berlin (2014)MATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Applied Mathematics and IUMAUniversity of ZaragozaZaragozaSpain

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