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Original Articles

A parameter-uniform hybrid finite difference scheme for singularly perturbed system of parabolic convection-diffusion problems

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Pages 875-905 | Received 02 Feb 2018, Accepted 22 Feb 2019, Published online: 03 Apr 2019

References

  • S. Bellew and E. O'Riordan, A parameter robust numerical method for a system of two singularly perturbed convection-diffusion equations, Appl. Numer. Math. 51(2–3) (2004), pp. 171–186. doi: 10.1016/j.apnum.2004.05.006
  • Z. Cen, Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations, Int. J. Comput. Math. 82(2) (2005), pp. 177–192. doi: 10.1080/0020716042000301798
  • Z. Cen, A. Xu, and A. Le, A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems, J. Comput. Appl. Math. 234(12) (2010), pp. 3445–3457. doi: 10.1016/j.cam.2010.05.006
  • C. Clavero, J.C. Jorge, F. Lisbona, and G.I. Shishkin, A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion problems, Appl. Numer. Math. 27(3) (1998), pp. 211–231. doi: 10.1016/S0168-9274(98)00014-2
  • C. Clavero, J.L. Gracia, and M. Stynes, A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems, J. Comput. Appl. Math. 235(17) (2011), pp. 5240–5248. doi: 10.1016/j.cam.2011.05.025
  • P. Das and S. Natesan, Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction-diffusion boundary-value problems, Appl. Math. Comput. 249 (2014), pp. 265–277.
  • A. Das and S. Natesan, Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh, Appl. Math. Comput. 271 (2015), pp. 168–186.
  • V. Franklin, M. Paramasivam, J.J.H. Miller, and S. Valarmathi, Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear parabolic system, Int. J. Numer. Anal. Model. 10(1) (2013), pp. 178–202.
  • J.L. Gracia and F.J. Lisbona, A uniformly convergent scheme for a system of reaction-diffusion equations, J. Comput. Appl. Math. 206(1) (2007), pp. 1–16. doi: 10.1016/j.cam.2006.06.005
  • J.L. Gracia, F.J. Lisbona, and E. O'Riordan, A coupled system of singularly perturbed parabolic reaction-diffusion equations, Adv. Comput. Math. 32(1) (2010), pp. 43–61. doi: 10.1007/s10444-008-9086-3
  • R.B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32(144) (1978), pp. 1025–1039. doi: 10.1090/S0025-5718-1978-0483484-9
  • O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
  • T. Linß, Analysis of a system of singularly perturbed convection-diffusion equations with strong coupling, SIAM J. Numer. Anal. 47(3) (2009), pp. 1847–1862. doi: 10.1137/070683970
  • J.J.H. Miller, E. O'Riordan, and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 2012.
  • K. Mukherjee and S. Natesan, Parameter-uniform hybrid numerical scheme for time-dependent convection-dominated initial-boundary-value problems, Computing 84(3–4) (2009), pp. 209–230. doi: 10.1007/s00607-009-0030-2
  • K. Mukherjee and S. Natesan, ϵ-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with interior layers, Numer. Algor. 58 (2011), pp. 103–141. doi: 10.1007/s11075-011-9449-6
  • C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
  • R.M. Priyadharshini, N. Ramanujam, and A. Tamilselvan, Hybrid difference schemes for a system of singularly perturbed convection-diffusion equations, J. Appl. Math. Inform. 27(5–6) (2009), pp. 1001–1015.
  • S.C.S. Rao and V. Srivastava, Parameter-robust numerical method for time-dependent weakly coupled linear system of singularly perturbed convection-diffusion equations, Differ. Equ. Dyn. Syst. 25(2) (2017), pp. 301–325. doi: 10.1007/s12591-016-0282-1
  • H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2008.
  • M.K. Singh and S. Natesan, Richardson extrapolation technique for singularly perturbed system of parabolic partial differential equations with exponential boundary layers, Appl. Math. Comput. 333 (2018), pp. 254–275.
  • M. Stynes and H.-G. Roos, The midpoint upwind scheme, Appl. Numer. Math. 23(3) (1997), pp. 361–374. doi: 10.1016/S0168-9274(96)00071-2
  • M. Stynes and L. Tobiska, A finite difference analysis of a streamline diffusion method on a Shishkin mesh, Numer. Algorithms. 18(3–4) (1998), pp. 337–360. doi: 10.1023/A:1019185802623

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