Monotone second order parameter-robust numerical methods for singularly perturbed differential equations can be designed using the principles of defect-correction. However, the proofs of second order parameter-uniform convergence can be difficult, even in one dimension. In this paper, we examine a variant of the standard defect-correction scheme. This variant is proposed in order to simplify the analysis of a defect-correction method in the case of a one dimensional convection–diffusion problem. Numerical results are presented to validate the theoretical results.
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V.B. Andréiev and I.A. Savin, The uniform convergence with respect to a small parameter of A.A. Samarskii's monotone scheme and its modification, Comput. Math. Math. Phys. 35 (1995) 581–591.
V.B. Andréiev and N.V. Kopteva, A study of difference schemes with the first derivative approximated by a central difference ratio, Comput. Math. Math. Phys. 36(8) (1996) 1065–1078.
V.B. Andréiev and N.V. Kopteva, On the convergence, uniform with respect to a small parameter, of monotone three-point finite-difference approximations, Diff. Equ. 34(7) (1998) 921–928.
V.B. Andréiev and N.V. Kopteva, Uniform with respect to a small parameter convergence of difference schemes for a convection–diffusion problem, in Analytical and Computational Methods for Convection-Dominated and Singularly Perturbed Problems, eds. J.J.H. Miller, G.I. Shishkin and L. Vulkov (Nova Science, New York, 2000) pp. 133–139.
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 (2004) 171–186.
C. Clavero, J.L. Gracia and F. Lisbona, High order methods on Shishkin meshes for singular perturbation problems of convection–diffusion type, Numer. Algorithms 22 (1999) 73–97.
V. Ervin and W. Layton, An analysis of a defect-correction method for a model convection–diffusion equation, SIAM J. Numer. Anal. 26(1) (1989) 169–179.
P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O'Riordan and G.I. Shishkin, Robust Computational Techniques for Boundary Layers (Chapman and Hall/CRC, Boca Raton, Florida, 2000).
A. Frohner, T. Linβ and H.-G. Roos, Defect correction on Shishkin-type meshes, Numer. Algorithms 26 (2001) 281–299.
P.W. Hemker, An accurate method without directional bias for the numerical solution of a 2-D elliptic singular perturbation problem, in Theory and Applications of Singular Perturbations. Lecture Notes in Mathematics 942, eds. W. Eckhaus and E.M. de Jager (Springer, Berlin Heidelberg New York, 1982).
P.W. Hemker, G.I. Shishkin and L.P. Shishkina, The use of defect correction for the solution of parabolic singular perturbation problems, ZAMM 77(1) (1997) 59–74.
P.W. Hemker, G.I. Shishkin and L.P. Shishkina, \(\varepsilon\)-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems, IMA J. Numer. Anal. 20(1) (2000) 99–121.
P.W. Hemker, G.I. Shishkin and L.P. Shishkina, High-order time-accurate schemes for parabolic singular perturbation problems with convection, Russ. J. Numer. Anal. Math. Model. 17(1) (2002) 1–24.
P.W. Hemker, G.I. Shishkin and L.P. Shishkina, Novel defect correction high-order, in space and time, accurate schemes for parabolic singularly perturbed convection–diffusion problems, Comput. Methods Appl. Math. 3(3) (2003) 387–404.
N.V. Kopteva, A study of certain difference schemes for convection–diffusion equations, Diplom Thesis, Moscow State University, 1993.
N.V. Kopteva, Uniform convergence with respect to a small parameter of a four-point scheme for the one-dimensional stationary convection–diffusion equation, Diff. Equ. 32(7) (1996) 951–957.
T. Linβ and M. Stynes, Numerical methods on Shishkin meshes for linear convection–diffusion problems, Comput. Methods Appl. Mech. Eng. 190 (2001) 3527–3542.
J.J.H. Miller, E. O'Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems (World-Scientific, Singapore, 1996).
J.J.H. Miller, E. O'Riordan, G.I. Shishkin and L.P. Shishkina, Fitted mesh methods for problems with parabolic boundary layers, Math. Proc. R. Ir. Acad. 98A(2) (1998) 173–190.
H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion and Flow Problems (Springer, Berlin Heidelberg New York, 1996).
G.I. Shishkin, Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations (Russian Academy of Sciences, Ural section, Ekaterinburg, 1992) (In Russian).
G.I. Shishkin, Method of improving the accuracy of the approximate solutions to singularly perturbed equations by defect correction, Russ. J. Numer. Anal. Math. Model. 11(6) (1996) 539–557.
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Communicated by M. Gasca
*This research was partially supported by the project MEC/FEDER MTM 2004-019051 and the grant EUROPA XXI of the Caja de Ahorros de la Inmaculada.
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Gracia, J.L., O'Riordan, E. A defect–correction parameter-uniform numerical method for a singularly perturbed convection–diffusion problem in one dimension. Numer Algor 41, 359–385 (2006). https://doi.org/10.1007/s11075-006-9021-y
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DOI: https://doi.org/10.1007/s11075-006-9021-y