## Taiwanese Journal of Mathematics

### New Finite Difference Methods for Singularly Perturbed Convection-diffusion Equations

#### Abstract

In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered. By taking account of infinite terms in the Taylor's expansions and using the triangle function theorem, we construct a series of NFD schemes for the one-dimensional problems firstly and derive the error estimates as well. Then, applying the ADI technique, the idea is extended to two dimensional equations. Besides no numerical oscillation, there are mainly three advantages for the proposed methods: one is that the schemes can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter for 1D equations; Secondly, no matter which convergence order the scheme is, the generated linear systems have diagonal structures; Thirdly, the methods are easily expanded to the special mesh technique such as Shishkin mesh. Some numerical experiments are shown to verify the prediction.

#### Article information

Source
Taiwanese J. Math., Volume 22, Number 4 (2018), 949-978.

Dates
Revised: 2 October 2017
Accepted: 16 October 2017
First available in Project Euclid: 26 October 2017

https://projecteuclid.org/euclid.twjm/1508983229

Digital Object Identifier
doi:10.11650/tjm/171002

Mathematical Reviews number (MathSciNet)
MR3830829

Zentralblatt MATH identifier
06965405

#### Citation

He, Xuefei; Wang, Kun. New Finite Difference Methods for Singularly Perturbed Convection-diffusion Equations. Taiwanese J. Math. 22 (2018), no. 4, 949--978. doi:10.11650/tjm/171002. https://projecteuclid.org/euclid.twjm/1508983229

#### References

• O. Axelsson and M. Nikolova, Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method, Computing 58 (1997), no. 1, 1–30.
• C. E. Baumann and J. T. Oden, A discontinuous $hp$ finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (1999), no. 3-4, 311–341.
• P. H. Chiu and T. W. H. Sheu, On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation, J. Comput. Phys. 228 (2009), no. 10, 3640–3655.
• P. C. Chu and C. Fan, A three-point combined compact difference scheme, J. Comput. Phys. 140 (1998), no. 2, 370–399.
• ––––, A three-point sixth-order nonuniform combined compact difference scheme, J. Comput. Phys. 148 (1999), no. 2, 663–674.
• R. Frank, The method of iterated defect-correction and its application to two-point boundary value problems I, Numer. Math. 25 (1976), no. 4, 409–419.
• R. Frank and C. W. Ueberhuber, Iterated defect correction for differential equations I: Theoretical results, Computing 20 (1978), no. 3, 207–228.
• L. Ge and J. Zhang, High accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids, J. Comput. Phys. 171 (2001), no. 2, 560–578.
• M. M. Gupta, J. Kouatchou and J. Zhang, A compact multigrid solver for convection-diffusion equations, J. Comput. Phys. 132 (1997), no. 1, 123–129.
• P. S. Huyakorn, Solution of steady-state, convective transport equation using an upwind finite element scheme, Appl. Math. Model. 1 (1977), no. 4, 187–195.
• R. D. Lazarov, I. D. Mishev and P. S. Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal. 33 (1996), no. 1, 31–55.
• J. Li, Optimal uniform convergence analysis for a singularly perturbed quasilinear reaction-diffusion problem, J. Numer. Math. 12 (2004), no. 1, 39–54.
• J. Li and Y. Chen, Uniform convergence analysis for singularly perturbed elliptic problems with parabolic layers, Numer. Math. Theory Methods Appl. 1 (2008), no. 2, 138–149.
• D. Liang and W. Zhao, A high-order upwind method for the convection-diffusion problem, Comput. Methods Appl. Mech. Engrg. 147 (1997), no. 1-2, 105–115.
• T. Linß, Layer-adapted meshes for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 9-10, 1061–1105.
• T. Linß and M. Stynes, Numerical methods on Shishkin meshes for linear convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 28, 3527–3542.
• V. Nassehi and S. A. King, Finite element methods for the convection diffusion equation, J. Eng. 4 (1991), no. 3, 93–100.
• M. C. Natividad and M. Stynes, Richardson extrapolation for a convection-diffusion problem using a Shishkin mesh, Appl. Numer. Math. 45 (2003), no. 2-3, 315–329.
• A. C. R. Pillai, Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, Internat. J. Numer. Methods Fluids 37 (2001), no. 1, 87–106.
• H.-G. Roos, Robust numerical methods for singularly perturbed differential equations: a survey covering 2008–2012, ISRN Appl. Math. 2012 (2012), Art. ID 379547, 30 pp.
• H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-diffusion-reaction and flow problems, Second edition, Springer Series in Computational Mathematics 24, Springer-Verlag, Berlin, 2008.
• Z. Z. Sun, Numerical Methods for Partical Difference Equations, (Chinese) Second edition, Science Press, Beijing, 2012.
• H. Sun and J. Zhang, A high-order finite difference discretization strategy based on extrapolation for convection diffusion equations, Numer. Methods Partial Differential Equations 20 (2004), no. 1, 18–32.
• Z. F. Tian and S. Q. Dai, High-order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys. 220 (2007), no. 2, 952–974.
• M. Vlasak and H.-G. Roos, An optimal uniform a priori error estimate for an unsteady singularly perturbed problem, Int. J. Numer. Anal. Model. 11 (2014), no. 1, 24–33.
• R. Vulanović and L. Teofanov, On the singularly perturbed semilinear reaction-diffusion problem and its numerical solution, Int. J. Numer. Anal. Model. 13 (2016), no. 1, 41–57.
• K. Wang and Y. S. Wong, Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model. 11 (2014), no. 4, 787–815.
• K. Wang, Y. S. Wong and J. Deng, Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates, Commun. Comput. Phys. 17 (2015), no. 3, 779–807.
• J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusion equations, Math. Comp. 68 (1999), no. 228, 1429–1446.
• S. Zhai and X. Feng, A new coupled high-order compact method for the three-dimensional nonlinear biharmonic equations, Int. J. Comput Math. 91 (2014), no. 10, 2307–2325.
• S. Zhai, X. Feng and Y. He, An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation, J. Comput. Phys. 269 (2014), 138–155.