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http://dx.doi.org/10.14317/jami.2018.181

SCHWARZ METHOD FOR SINGULARLY PERTURBED SECOND ORDER CONVECTION-DIFFUSION EQUATIONS  

ROJA, J. CHRISTY (Department of Mathematics, St.Joseph's college)
TAMILSELVAN, A. (Department of Mathematics, Bharathidasan University)
Publication Information
Journal of applied mathematics & informatics / v.36, no.3_4, 2018 , pp. 181-203 More about this Journal
Abstract
In this paper, we have constructed an overlapping Schwarz method for singularly perturbed second order convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations which converge in the maximum norm to the exact solution. When appropriate subdomains are used, the numerical approximations generated from the method are shown to be first order convergent. Furthermore it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantages of this method used with the proposed scheme is it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.
Keywords
Singularly perturbed problems; Convection-diffusion equations; Schwarz method; Hybrid difference scheme;
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1 S. Kumar and S.C.S. Rao, A robust overlapping Schwarz domain decomposition algorithm for time-dependent singularly perturbed reactiondiffusion problems, J. Comput. Appl. Math. 261(2014), 127-138.   DOI
2 H. MacMullen, J.J.H. Miller, E. O'Riordan and G.I Shishkin, A second order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers, J.Comput. Appl. Math. 130(2001), No.1-2, 231-244.   DOI
3 H. MacMullen, E. O'Riordan and G.I Shishkin, The convergence of classical Schwarz methods applied to convection-diffusion problems with regular boundary layers. Appl. Num. Math. 43(2002), 297-313.   DOI
4 H. MacMullen, J.J.H. Miller, E. O'Riordan and G.I Shishkin, Schwarz iterative method for convection-diffusion problems with boundary layers in: J.J.H. Miller, G.I. Shishkin, L.G. Vulkov,(Eds.) Analytical and numerical methods for convection-dominated and singularly perturbed problems, Nova science publishers, New York, (2000), 213-218.
5 J.J.H. Miller, E. O'Riordan and G.I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, 1996.
6 M. Garbey, A Schwarz alternating procedure for singular perturbation problems, SIAM, J. Sci. Comput. 17(1996), 1175-1201.   DOI
7 R. Mythili Priyadharshini, N. Ramanujam and A. Tamilselvan, Hybrid difference schemes for a system of singularly perturbed convection-diffusion equations, J. Appl. Math. & Informatics 27(2009), No.(5-6), 1001-1015.
8 E. O'Riordan, J.L. Gracia and M.L. Pickett, A parameter robust second order numerical method for a singularly perturbed two-parameter problem, Appl. Num. Math. 56(2006), 962-980.   DOI
9 S.C.S. Rao and S. Kumar, An almost fourth order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reactiondiffusion equations, J. Comput. Appl. Math. 235(2011), 33423354.   DOI
10 M. Stephens and N. Madden, A parameter-uniform Schwarz method for a coupled system of reaction-diffusion equations, J. Comput. Appl. Math. 230(2009), 360-370.   DOI
11 M. Stephens and N. Madden, A Schwarz technique for a system of reaction diffusion equations with differing parameters, in: A. Hegarty, N. Kopteva, E. ORiordan, M. Stynes (Eds.), BAIL 2008. Boundary and Interior Layers, in: Lecture Notes in Computational Science and Engineering 69(2009), Springer, Berlin, Heidelberg, 247-255.
12 Sunil Kumar and Mukesh Kumar, An analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems, J. Comput. Appl. Math. 281(2015), 250262   DOI
13 M. Stynes and H.G. Roos, The mid-point upwind scheme, Appl. Num. Math. 23(1997), 362-371.
14 M. Garbey and H.G. Kaper, Heterogeneous domain decomposition for singularly perturbed elliptic boundary value problems, SIAM, J. Num. Analy. 34(1997), 1513-1544.   DOI
15 T.P Mathew, Uniform convergence of the Schwarz alternating method for solving singularly perturbed advection-diffusion equations, SIAM, J.Num. Analy. 35(1998), 1663-1683.   DOI