Geodesic
Parameter-uniform numerical methods for a class of parameterized singular perturbation problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 2, pp. 213-228.

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In this article, a weighted finite difference scheme is proposed for solving a class of parameterized singularly perturbed problems (SPPs). Depending upon the choice of the weight parameter, the scheme is automatically transformed from the backward Euler scheme to a monotone hybrid scheme. Three kinds of nonuniform grids are considered: a standard Shishkin mesh, a Bakhavalov–Shishkin mesh, and an adaptive grid. The methods are shown to be uniformly convergent with respect to the perturbation parameter for all three types of meshes. The rate of convergence is of first order for the backward Euler scheme and of second order for the monotone hybrid scheme. Furthermore, the proposed method is extended to a parameterized problem with mixed type boundary conditions and is shown to be uniformly convergent. Numerical experiments are carried out to show the efficiency of the proposed schemes, which indicate that the estimates are optimal. 
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D. Shakti; J. Mohapatra. Parameter-uniform numerical methods for a class of parameterized singular perturbation problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 2, pp. 213-228. http://geodesic.mathdoc.fr/item/SJVM_2019_22_2_a7/

[1] Amiraliyev G.M., Duru H., “A note on a parameterized singular perturbation problem”, J. Comput. Appl. Math., 182 (2005), 233-242 | DOI | MR | Zbl

[2] Amiraliyev G.M., Kudu M., Duru H., “Uniform difference method for a parameterized singular perturbation problem”, Appl. Math. Comput., 175 (2006), 89-100 | MR | Zbl

[3] Beckett G., Mackenzie J.A., “Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem”, Appl. Numer. Math., 35:106 (2000), 87-109 | DOI | MR | Zbl

[4] de Boor C., “Good approximation by splines with variable knots”, Spline Functions and Approximation Theory, Proc. of the Symposium (University of Alberta, Edmonton, May 29–June 1, 1972), eds. A. Meir, A. Sharma, Birkhäuser, Basel, 1973 | MR

[5] Cen Z., “A second-order difference scheme for a parameterized singular perturbation problem”, J. Comput. Appl. Math., 221 (2008), 174-182 | DOI | MR | Zbl

[6] Das P., Natesan S., “Numerical solution of a system of singularly perturbed convection diffusion boundary value problems using mesh equidistribution technique”, Aust. J. Math. Anal. Appl., 10 (2013), 1-17 | MR | Zbl

[7] Das P., Natesan S., “Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction-diffusion boundary-value problems”, Appl. Math. Comput., 249 (2014), 265-277 | MR | Zbl

[8] Chen Y., “Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid”, Adv. Comput. Math., 24 (2006), 197-212 | DOI | MR | Zbl

[9] Farrell P.A., Hegarty A.F., Miller J.M., O'Riordan E., Shishkin G.I., Robust Computational Techniques for Boundary Layers, Chapman Hall/CRC Press, Boca Raton, FL, 2000 | MR | Zbl

[10] Feckan M., “Parametrized singularly perturbed boundary value problems”, J. Math. Anal. Appl., 188 (1994), 426-435 | DOI | MR | Zbl

[11] Jankowski T., Lakshmikantham V., “Monotone iterations for differential equations with a parameter”, J. Appl. Math. Stoch. Anal., 10:3 (1997), 273-278 | DOI | MR | Zbl

[12] Kellogg R.B., Tsan A., “Analysis of some difference approximations for a singular perturbation problem without turning points”, Math. Comput., 32 (1978), 1025-1039 | DOI | MR | Zbl

[13] Kopteva N., Stynes M., “A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem”, SIAM J. Numer. Anal., 39 (2001), 1446-1467 | DOI | MR | Zbl

[14] Kopteva N., Madden N., Stynes M., “Grid equidistribution for reaction-diffusion problems in one dimension”, Numer. Algorithms., 40 (2005), 305-322 | DOI | MR | Zbl

[15] Liu X., McRae F.A., “A Monotone iterative method for boundary value problems of parametric differential equations”, J. Appl. Math. Stoch. Anal., 14:2 (2001), 183-187 | DOI | MR | Zbl

[16] LinßT., “Sufficient conditions for uniform convergence on layer-adapted grids”, Appl. Numer. Math., 37 (2001), 241-255 | DOI | MR | Zbl

[17] Liseikin V.D., Grid Generation Methods, Springer, Berlin, 1999 | MR | Zbl

[18] Liseikin V.D., Layer Resolving Grids and Transformations for Singular Perturbation Problems, VSP, 2001

[19] Mackenzie J., “Uniform convergence analysis of an upwind finite-difference approximation of a convection-diffusion boundary value problem on an adaptive grid”, IMA J. Numer. Anal., 19 (1999), 233-249 | DOI | MR | Zbl

[20] Miller J.J.H., O'Riordan E., Shishkin G.I., Fitted Numerical Methods for Singular Perturbation Problems, Revised edition, World Scientific, Singapore, 2012 | MR | Zbl

[21] Mohapatra J., Natesan S. Parameter-uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution, Comput. Math. Appl., 60 (2010), 1924-1939 | DOI | MR | Zbl

[22] Pomentale T., “A constructive theorem of existence and uniqueness for problem $y'=f(x, y, \lambda)$, $y(a)=\alpha$, $y(b)=\beta$”, Z. Angrew. Math. Mech., 56:8 (1976), 387-388 | DOI | MR | Zbl

[23] Ronto M., Csikos-Marinets T., “On the investigation of some non-linear boundary value problems with parameters”, Math. Notes (Miskolc), 1:2 (2000), 157-166 | DOI | MR | Zbl

[24] Roos H.G., Stynes M., Tobiska L., Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 2008 | MR | Zbl

[25] Shakti D., Mohapatra J., “A second order numerical method for a class of parameterized singular perturbation problems on adaptive grid”, Nonlinear Engineering, 6:3 (2017), 221-228 | DOI

[26] Turkyilmazoglu M., “Analytic approximate solutions of parameterized unperturbed and singularly perturbed boundary value problems”, Appl. Math. Model., 35 (2011), 3879-3886 | DOI | MR | Zbl

[27] Xie F., Wang J., Zhang W., He M., “A novel method for a class of parameterized singularly perturbed boundary value problems”, J. Comput. Appl. Math., 213 (2008), 258-267 | DOI | MR | Zbl

[28] Xu X., Huang W., Russell R.D., Williams J.F., “Convergence of de Boor's algorithm for the generation of equidistributing meshes”, IMA J. Numer. Anal., 31 (2011), 580-596 | DOI | MR | Zbl