High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type

Numerical Algorithms

Volumn 22, Issue 1, 1999, Pages 73-97

  Clavero, C ^a   Gracia, J L ^a   Lisbona, F ^a  

^a UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

High order; Shishkin mesh; Singular perturbation; Uniform convergence

Indexed keywords

EID: 0033435826     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1019150606200     Document Type: Article

References (21)

1. V.B. Andréiev and N.V. Kopteva, A study of difference schemes with the first derivate approximated by a central difference ratio, Comput. Math. Math. Phys. 36 (1996) 1065-1078.
2. A.S. Bakhvalov, On the optimization of methods for solving boundary value problems with a boundary layer, U.S.S.R. Comput. Math. Math. Phys. 9 (1969) 139-166.
3. C. Clavero, F. Lisbona and J.J.H. Miller, Uniform convergence of arbitrary order on nonuniform meshes for a singularly perturbed boundary value problem, J. Comput. Appl. Math. 59 (1995) 155-171.
4. E.C. Gartland, Uniform high-order difference schemes for a singularly perturbed two point boundary value problem, Math. Comp. 48 (1987) 551-564.
5. E.C. Gartland, Graded-mesh difference schemes for singularly perturbed two-point boundary value problems, Math. Comp. 51 (1988) 631-657.
6. D. Herceg, Uniform fourth order difference scheme for a singular perturbation problem, Numer. Math. 56 (1990) 675-693.
7. R.B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 (1978) 1025-1039.
8. N.V. Kopteva, On the convergence, uniform with respect to a small parameter, of a four-point scheme for a one-dimensional stationary convection-diffusion equation, Differential Equations 32 (1997) 958-964.
9. R.E. Lynch and J.R. Rice, A high-order difference method for differential equations, Math Comp. 34 (1980) 333-372.
10. J.M. Melenk and C. Schwab, The hp streamline diffusion finite element method for convection dominated problems in one space dimension, Research Report No. 98-10, Seminar für Angewandte Mathematik, Zurich (October 1998).
11. J.J.H. Miller, E. O'Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (World Scientific, Singapore, 1996).
12. M. Nikolova and O. Axelsson, Uniform in ε convergence of defect-correction method for convection-diffusion problems, Report No. 9905, Department of Mathematics, University of Nijmegen (February 1999).
13. H.G. Roos, Layer-adapted grids for singular perturbation problems, ZAMM 78 (1998) 291-309.
14. H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer, Berlin, 1996).
15. G.I. Shishkin, A difference scheme on a non-uniform mesh for a differential equation with a small parameter in highest derivative, U.S.S.R. Comput. Math. Math. Phys. 23 (1983) 59-66.
16. G.I. Shishkin, Grid approximation of singularly perturbed boundary value problems with convective terms, Soviet J. Numer. Anal. Math. Modeling 5 (1990) 173-187.
17. M. Stynes and H.G. Roos, The midpoint upwind scheme, Appl. Numer. Math. 23 (1997) 361-374.
18. M. Stynes and L. Tobiska, A finite difference analysis of a streamline diffusion method on a Shishkin mesh, to appear in Numer. Algorithms.
19. R. Vulanovic, On a numerical solution of a type of singularly perturbed problem by using a special discretization mesh, Univ. Novom Sadu Zb. Rad. Prirod., Mat. Fak Ser. Mat. 13 (1983) 187-201.
20. R. Vulanovic, Non-equidistant generalizations of the Gushchin-Shennikov scheme, ZAMM 67 (1987) 625-632.
21. P. Wesseling, Uniform convergence of discretization error for a singular perturbation problem, Numer. Methods for Partial Differential Equations 12 (1996) 657-671.