# Opposing flows in a one dimensional convection-diffusion problem

Open Mathematics (2012)

- Volume: 10, Issue: 1, page 85-100
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topEugene O’Riordan. "Opposing flows in a one dimensional convection-diffusion problem." Open Mathematics 10.1 (2012): 85-100. <http://eudml.org/doc/269267>.

@article{EugeneO2012,

abstract = {In this paper, we examine a particular class of singularly perturbed convection-diffusion problems with a discontinuous coefficient of the convective term. The presence of a discontinuous convective coefficient generates a solution which mimics flow moving in opposing directions either side of some flow source. A particular transmission condition is imposed to ensure that the differential operator is stable. A piecewise-uniform Shishkin mesh is combined with a monotone finite difference operator to construct a parameter-uniform numerical method for this class of singularly perturbed problems.},

author = {Eugene O’Riordan},

journal = {Open Mathematics},

keywords = {Singularly perturbed; Shishkin mesh; Parameter-uniform convergence; convection-diffusion problem; singular perturbation; parameter-uniform convergence; second-order two-point boundary value problems; discontinuous coefficient; stability; error estimate; numerical example},

language = {eng},

number = {1},

pages = {85-100},

title = {Opposing flows in a one dimensional convection-diffusion problem},

url = {http://eudml.org/doc/269267},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Eugene O’Riordan

TI - Opposing flows in a one dimensional convection-diffusion problem

JO - Open Mathematics

PY - 2012

VL - 10

IS - 1

SP - 85

EP - 100

AB - In this paper, we examine a particular class of singularly perturbed convection-diffusion problems with a discontinuous coefficient of the convective term. The presence of a discontinuous convective coefficient generates a solution which mimics flow moving in opposing directions either side of some flow source. A particular transmission condition is imposed to ensure that the differential operator is stable. A piecewise-uniform Shishkin mesh is combined with a monotone finite difference operator to construct a parameter-uniform numerical method for this class of singularly perturbed problems.

LA - eng

KW - Singularly perturbed; Shishkin mesh; Parameter-uniform convergence; convection-diffusion problem; singular perturbation; parameter-uniform convergence; second-order two-point boundary value problems; discontinuous coefficient; stability; error estimate; numerical example

UR - http://eudml.org/doc/269267

ER -

## References

top- [1] Brayanov I.A., Uniformly convergent finite volume difference scheme for 2D convection-dominated problem with discontinuous coefficients, Appl. Math. Comput., 2005, 163(2), 645–665 http://dx.doi.org/10.1016/j.amc.2004.04.007 Zbl1069.65113
- [2] Dunne R.K., O’Riordan E., Interior layers arising in linear singularly perturbed differential equations with discontinuous coefficients, In: Proceedings of the Fourth International Conference on Finite Difference Methods: Theory and Applications, Lozenetz, August 26–29, 2006, Rousse University, Bulgaria, 2007, 29–38
- [3] de Falco C., O’Riordan E., Interior layers in a reaction-diffusion equation with a discontinuous diffusion coefficient, Int. J. Numer. Anal. Model., 2010, 7(3), 444–461 Zbl1202.65096
- [4] de Falco C., O’Riordan E., A parameter robust Petrov-Galerkin scheme for advection-diffusion-reaction equations, Numer. Algorithms, 2011, 56(1), 107–127 http://dx.doi.org/10.1007/s11075-010-9376-y Zbl1208.65171
- [5] Farrell P.A., Hegarty A.F., Miller J.J.H., O’Riordan E., Shishkin G.I., Robust Computational Techniques for Boundary Layers, Appl. Math. (Boca Raton), 16, Chapman & Hall/CRC Press, Boca Raton, 2000 Zbl0964.65083
- [6] Farrell P.A., Hegarty A.F., Miller J.J.H., O’Riordan E., Shishkin G.I., Singularly perturbed convection-diffusion problems with boundary and weak interior layers, J. Comput. Appl. Math., 2004, 166(1), 133–151 http://dx.doi.org/10.1016/j.cam.2003.09.033 Zbl1041.65059
- [7] Farrell P.A., Hegarty A.F., Miller J.J.H., O’Riordan E., Shishkin G.I., Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient, Math. Comput. Modelling, 2004, 40(11–12), 1375–1392 http://dx.doi.org/10.1016/j.mcm.2005.01.025 Zbl1075.65100
- [8] Linß T., Finite difference schemes for convection-diffusion problems with a concentrated source and a discontinuous convection field, Comput. Methods Appl. Math., 2002, 2(1), 41–49 Zbl0995.65078
- [9] O’Riordan E., Pickett M.L., Shishkin G.I., Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems, Math. Comp., 2006, 75(255), 1135–1154 http://dx.doi.org/10.1090/S0025-5718-06-01846-1 Zbl1098.65091
- [10] O’Riordan E., Shishkin G.I., Singularly perturbed parabolic problems with non-smooth data, J. Comput. Appl. Math., 2004, 166(1), 233–245 http://dx.doi.org/10.1016/j.cam.2003.09.025 Zbl1041.65071
- [11] Roos H.-G., Stynes M., Tobiska L., Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd ed., Springer Ser. Comput. Math., 24, Springer, Berlin, 2008 Zbl1155.65087
- [12] Shishkin G.I., A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous initial condition, Soviet Math. Dokl., 1988, 37(3), 792–796 Zbl0662.65087
- [13] Shishkin G.I., A difference scheme for a singularly perturbed parabolic equation with discontinuous coefficients and concentrated factors, U.S.S.R. Comput. Math. and Math. Phys., 1989, 29(5), 9–19 http://dx.doi.org/10.1016/0041-5553(89)90173-0
- [14] Shishkin G.I., Approximation of singularly perturbed parabolic reaction-diffusion equations with nonsmooth data, Comput. Methods Appl. Math., 2001, 1(3), 298–315 Zbl1098.65508