Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T08:51:20.170Z Has data issue: false hasContentIssue false
  • English
  • Français

An Adaptive Grid Method for Singularly Perturbed Time-Dependent Convection-Diffusion Problems

Part of: Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  02 November 2016

Yanping Chen  and
Li-Bin Liu
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Li-Bin Liu
Affiliation:
School of Mathematics and Computer, Chizhou University, Chizhou, Anhui 247000, China
*
*Corresponding author. Email address:[email protected] (Y. Chen)
Get access

Abstract

In this paper, we study the numerical solution of singularly perturbed time-dependent convection-diffusion problems. To solve these problems, the backward Euler method is first applied to discretize the time derivative on a uniform mesh, and the classical upwind finite difference scheme is used to approximate the spatial derivative on an arbitrary nonuniform grid. Then, in order to obtain an adaptive grid for all temporal levels, we construct a positive monitor function, which is similar to the arc-length monitor function. Furthermore, the ε-uniform convergence of the fully discrete scheme is derived for the numerical solution. Finally, some numerical results are given to support our theoretical results.

Keywords

Uniform convergencesingularly perturbedconvection-diffusion problemsadaptive gridan upwind finite difference scheme

MSC classification

Secondary: 65L10: Boundary value problems 65L12: Finite difference methods 65L50: Mesh generation and refinement
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1340 - 1358
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Roos, H. G., Stynes, M., Tobiska, L., Numerical methods for singularly perturbed differential equations. Spring, Berlin, 1996.Google Scholar
[2] Miller, J. J. H., O’Riordan, E., Shishkin, G.I., Fitted numercial methods for singularly perturbation problems, World Scientific, Singapore, 1996.Google Scholar
[3] Clavero, C., Jorge, J. C., Lisbona, F., A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems, Journal of Computational and Applied Mathematics, 154 (2003) 415–429.Google Scholar
[4] Mukherjee, K., Natesan, S., Parameter-uniform hybird numerical scheme for time-dependent convection-dominated initial-boundary-value problems, Computing, 84 (2009) 209–230.CrossRefGoogle Scholar
[5] Mukherjee, K., Natesan, S., Richardson extrapolation technique for singularly perturbed parabolic convection-diffsion problems, Computing, 92 (2011) 1–32.Google Scholar
[6] Kadalbajoo, M. K., Awasthi, A., A parameter uniform difference scheme for singularly perturbed parabolic problemin one space dimension, Applied Mathematicas and Computation, 183 (2006) 42–60.Google Scholar
[7] Clavero, C., Gracia, J. L., Stynes, M., A simple analysis of a hybird numericalmethod for time-dependent convection-diffusion problems, Journal of Computational and Applied Mathematics, 235 (2011) 5240–5248.Google Scholar
[8] Gupta, V., Kadalbajoo, M. K., A layer adaptive B-spline collocation method for singularly perutrbed one-dimensional parabolic problem with a boundary turning point, Numerical Methods Partial Differential Equation, 27 (2011) 1143–1164.Google Scholar
[9] O’Riordan, E., Pickett, M. L., Shishkin, G. I., Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems, Mathematics of Computation, 75 (2006) 1135–1154.Google Scholar
[10] Jha, A., Kadalbajoo, M.K., A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems, Internation Journal of Computer Mathematics, 92 (2015) 1204–1221.CrossRefGoogle Scholar
[11] Gowrisankar, S., Natesan, S., Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids, International Journal of Computer Mathematics, 91 (2014) 553–577.Google Scholar
[12] Gowrisankar, S., Natesan, S., Robust numerical scheme for singularly perturbed convection-diffusion parabolic initial-boundary-value problems on equidistributed grids, Computer Physics Communications, 185 (2014) 2008–2019.Google Scholar
[13] Liu, L.-B., Chen, Y., A robust adaptive grid method for a system of two singularly perturbed Convection-diffusion equations with weak coupling, Journal of Scientific Computing, 61 (2014) 1–16.CrossRefGoogle Scholar
[14] Linß, T., Analysis of a system of singularly perturbed convection-diffusion equations with strong coupling, SIAM Journal on Numerical Analysis, 47 (2009) 1847–1862.CrossRefGoogle Scholar
[15] Cen, Z. D., Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations, International Journal of Computer Mathematics, 82 (2005) 177–192.Google Scholar
[16] Roos, H.-G., Reibiger, C., Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control, Numerical Mathematics: Theory, Methods and Applications, 4 (2011) 562–575.Google Scholar
[17] Linß, T., Dresden, Analysis of an upwind finite-difference scheme for a system of coupled singularly perturbed convection-diffusion equations, Computing, 79 (2007) 23–32.Google Scholar
[18] Qiu, Y., Sloan, D.M., Tang, T., Convergence analysis of an adaptive finite difference method for a singular perturbation problem, Journal of Computational and Applied Mathematics, 116 (2000) 121–143.Google Scholar
[19] Kopteva, N., Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM Journal on Numerical Analysis, 39 (2001) 423–441.CrossRefGoogle Scholar
[20] Kopteva, N., Stynes, M., A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem, SIAM Journal on Numerical Analysis, 39 (2001) 1446–1467.Google Scholar
[21] Chen, Y., Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution, Journal of Computational and Applied Mathematics, 159 (2003) 25–34.CrossRefGoogle Scholar
[22] Chen, Y., Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid, Advances in Computational Mathematics, 24 (2006) 197–212.CrossRefGoogle Scholar
[23] Mackenzie, J., Uniform convergence analysis of an upwind finite difference approximation of a convection-diffusion bondary value problem on an adaptive gird, IMA Journal of Numerical Analysis, 19 (1999) 233–249.CrossRefGoogle Scholar