Opposing flows in a one dimensional convection-diffusion problem

• Autorzy: Eugene O’Riordan
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Singularly perturbed; Shishkin mesh; Parameter-uniform convergence

Kategorie tematyczne

• 65M06: Finite difference methods
• 65L12: Finite difference methods
• 65L11: Singularly perturbed problems
• 65L20: Stability and convergence of numerical methods

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 1
• Strony: 85-100

Daty

wydano

2012-02-01

online

2011-12-09

Twórcy

autor

Eugene O’Riordan

• Dublin City University

Bibliografia

• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [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

Identyfikatory

DOI

10.2478/s11533-011-0121-0