A High Order Accurate Overlapping Domain Decomposition Method for Singularly Perturbed Reaction-Diffusion Systems

  • Joginder SinghEmail author
  • Sunil Kumar
  • Mukesh Kumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


In this work, we consider a coupled system of singularly perturbed reaction-diffusion equations (SPRDEs) with distinct small positive parameters, exhibiting overlapping boundary layers at both ends of the domain. In [4], the authors designed an overlapping domain decomposition method that gives almost second order accurate approximations to the solution of coupled systems of SPRDEs. High order methods are of great significance to the numerical community. To this end, for numerically solving the coupled systems of SPRDEs, we designed a high order accurate overlapping domain decomposition method via. defining an appropriate decomposition of the original domain and then considering a hybrid difference scheme on a uniform mesh on each subdomain. More precisely, the method gives almost fourth order accurate approximations to the solution of the problem, as compared to almost second order accurate approximations in [4]. Numerical results are given to demonstrate the effectiveness of the proposed method.


Singularly perturbed Coupled system Domain decomposition Reaction-diffusion 


  1. 1.
    Boglaev, I.: Domain decomposition for a parabolic convection-diffusion problem. Numer. Methods Partial Differ. Equ. 22, 1361–1378 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Clavero, C., Gracia, J.: High order methods for elliptic and time dependent reaction diffusion singularly perturbed problems. Appl. Math. Comput. 168, 1109–1127 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Kopteva, N., Pickett, M., Purtill, H.: A robust overlapping Schwarz method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions. Int. J. Numer. Anal. Model. 6, 680–695 (2009)MathSciNetGoogle Scholar
  4. 4.
    Kumar, S., Kumar, M.: An analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems. J. Comput. Appl. Math. 281, 250–262 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kumar, S., Rao, S.C.S.: A robust overlapping Schwarz domain decomposition algorithm for time-dependent singularly perturbed reaction-diffusion problems. J. Comput. Appl. Math. 261, 127–138 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Linß, T.: Layer-Adapted Meshes for Reaction-Convection-diffusion Problems. LNM, vol. 1985. Springer, Berlin (2010). Scholar
  7. 7.
    MacMullen, H., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers. J. Comput. Appl. Math. 130, 231–244 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Quarternoni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (1999)Google Scholar
  10. 10.
    Rao, S.C.S., Kumar, S.: An almost fourth order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reaction-diffusion equations. J. Comput. Appl. Math. 235, 3342–3354 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rao, S.C.S., Kumar, S.: Robust high order convergence of an overlapping Schwarz method for singularly perturbed semilinear reaction-diffusion problems. J. Comput. Math. 31, 509–521 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Stephens, M., Madden, N.: A parameter-uniform Schwarz method for a coupled system of reaction-diffusion equations. J. Comput. Appl. Math. 230, 360–370 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Stephens, M., Madden, N.: A Schwarz technique for a system of reaction diffusion equations with differing parameters. In: Hegarty, A., Kopteva, N., O’Riordan, E., Stynes, M. (eds.) BAIL 2008 - Boundary and Interior Layers. LNCSE, vol. 69, pp. 247–255. Springer, Berlin, Heidelberg (2009). Scholar
  14. 14.
    Stynes, M., Roos, H.: The midpoint upwind scheme. Appl. Numer. Math. 23, 361–374 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Toselli, A., Widlund, O.: Domain Decomposition Methods - Algorithms and Theory. SSCM. Springer, Heidelberg (2010)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndian Institute of Technology (BHU) VaranasiVaranasiIndia
  2. 2.Department of MathematicsCollege of CharlestonCharlestonUSA

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