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Parameter-uniform improved hybrid numerical scheme for singularly perturbed problems with interior layers

    Kaushik Mukherjee Affiliation

Abstract

In this paper, we consider a class of singularly perturbed convection-diffusion boundary-value problems with discontinuous convection coefficient which often occur as mathematical models for analyzing shock wave phenomena in gas dynamics. In general, interior layers appear in the solutions of this class of problems and this gives rise to difficulty while solving such problems using the classical numerical methods (standard central difference or standard upwind scheme) on uniform meshes when the perturbation parameter ε is small. To achieve better numerical approximation in solving this class of problems, we propose a new hybrid scheme utilizing a layer-resolving piecewise-uniform Shishkin mesh and the method is shown to be ε-uniformly stable. In addition to this, it is proved that the proposed numerical scheme is almost second-order uniformly convergent in the discrete supremum norm with respect to the parameter ε. Finally, extensive numerical experiments are conducted to support the theoretical results. Further, the numerical results obtained by the newly proposed scheme are also compared with the hybrid scheme developed in the paper [Z.Cen, Appl. Math. Comput., 169(1): 689-699, 2005]. It shows that the current hybrid scheme exhibits a significant improvement over the hybrid scheme developed by Cen, in terms of the parameter-uniform order of convergence.

Keyword : singularly perturbed boundary-value problem, interior layer, numerical scheme, piecewise-uniform Shishkin mesh, uniform convergence

How to Cite
Mukherjee, K. (2018). Parameter-uniform improved hybrid numerical scheme for singularly perturbed problems with interior layers. Mathematical Modelling and Analysis, 23(2), 167-189. https://doi.org/10.3846/mma.2018.011
Published
Apr 18, 2018
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19

References

[1] Z. Cen. A hybrid difference scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient. Applied Mathematics and Computation, 169(1):689–699, 2005. https://doi.org/10.1016/j.amc.2004.08.051

[2] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient. Mathematical and Computer Modelling, 40(11):1375–1392, 2004. https://doi.org/10.1016/j.mcm.2005.01.025

[3] P.A. Farrell, A.F. Hegarty, J.J.M. Miller, E. O’Riordan and G.I. Shishkin. Robust Computational Techniques for Boundary Layers. Chapman & Hall/CRC Press,2000.

[4] P.A. Farrell, E. O’Riordan and G.I. Shishkin. A class of singularly perturbed quasilinear differential equations with interior layers. Mathematics of Computation, 78(265):103–127, 2009. https://doi.org/10.1090/S0025-5718-08-02157-1

[5] M.K. Kadalbajoo and V. Gupta. A brief survey on numerical methods for solving singularly perturbed problems. Applied Mathematics and Computation, 217(8):3641–3716, 2010. https://doi.org/10.1016/j.amc.2010.09.059

[6] J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore, 1996.

[7] K. Mukherjee and S. Natesan. Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients. BIT Numerical Mathematics, 51(2):289–315,2011. https://doi.org/10.1007/s10543-010-0292-2

[8] K. Mukherjee and S. Natesan. ε-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with nterior layers. Numerical Algorithms, 58(1):103–141, 2011. https://doi.org/10.1007/s11075-011-9449-6

[9] E. O’Riordan and G.I. Shishkin. Singularly perturbed parabolic problems with non-smooth data. Journal of Computational and Applied Mathematics, 166(1):233–245, 2004. https://doi.org/10.1016/j.cam.2003.09.025

[10] R.M. Priyadharshini and N. Ramanujam. A hybrid difference scheme for singularly perturbed second order ordinary differential equations with discontinuous convection coefficient and mixed type boundary conditions. International Journal of Computational Methods, 05(04):575–593, 2008. https://doi.org/10.1142/S0219876208001637

[11] H.-G. Roos, M. Stynes and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, Heidelberg, 2008. https://doi.org/10.1007/978-3-540-34467-4

[12] V. Shanthi, N. Ramanujam and S. Natesan. Fitted mesh method for singularly perturbed reaction convection-diffusion problems with boundary and interior layers. Journal of Applied Mathematics and Computing, 22(1):49–65, 2006. https://doi.org/10.1007/BF02896460

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