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A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity

Published online by Cambridge University Press:  03 June 2015

Po-Wen Hsieh ,
Suh-Yuh Yang  and
Cheng-Shu You
Po-Wen Hsieh*
Affiliation:
Department of Applied Mathematics, Chung Yuan Christian University, Jhongli City, Taoyuan County 32023, Taiwan
Suh-Yuh Yang*
Affiliation:
Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan
Cheng-Shu You*
Affiliation:
Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan
*
Email: pwhsieh0209@gmail.com
Corresponding author. Email: syyang@math.ncu.edu.tw
Email: csdyou@gmail.com
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Abstract

This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivity ε. With a novel treatment for the reaction term, we first derive a difference scheme of accuracy O(εh2+εh2+h3) for the 1-D case. Using the alternating direction technique, we then extend the scheme to the 2-D case on a nine-point stencil. We apply the high-accuracy finite difference scheme to solve the 2-D steady incompressible Navier-Stokes equations in the stream function-vorticity formulation. Numerical examples are given to illustrate the effectiveness of the proposed difference scheme. Comparisons made with some high-order compact difference schemes show that the newly proposed scheme can achieve good accuracy with a better stability.

Keywords

65N0665N1265N1576M20Reaction-convection-diffusion equationincompressible Navier-Stokes equationsboundary layerinterior layerfinite difference scheme

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 5 , October 2014 , pp. 637 - 662
Copyright
Copyright © Global-Science Press 2014

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References

[1]Ding, H. and Zhang, Y., A new difference scheme with high accuracy and absolute stability for solving convection-diffusion equations, J. Comput. Appl. Math., 230 (2009), pp. 600–606.Google Scholar
[2]Elman, H. C., Silvester, D. J. and Wathen, A. J., Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, New York, 2005.Google Scholar
[3]Ghia, U., Ghia, K. N. and Shin, C. T., High Re-solution for incompressible Navier-Stokes equation and a multigrid method, J. Comput. Phys., 48 (1982), pp. 387–411.Google Scholar
[4]Gupta, M. M., High accuracy solutions of incompressible Navier-Stokes equations, J. Comput. Phys., 93 (1991), pp. 343–359.Google Scholar
[5]Gupta, M. M. and Kalita, J. C., A new paradigm for solving Navier-Stokes equations: streamfunction-velocity formulation, J. Comput. Phys., 207 (2005), pp. 52–68.Google Scholar
[6]Hsieh, P.-W. and Yang, S.-Y., Two new upwind difference schemes for a coupled system of convection-diffusion equations arising from the steady MHD duct flow problems, J. Comput. Phys., 229 (2010), pp. 9216–9234.Google Scholar
[7]Hsieh, P.-W. and Yang, S.-Y., A novel least-squaresfinite element method enriched with residualfree bubbles for solving convection-dominated problems, SIAM J. Sci. Comput., 32 (2010), pp. 2047–2073.Google Scholar
[8]Karaa, S. and Zhang, J., High order ADI method for solving unsteady convection-diffusion problems, J. Comput. Phys., 198 (2004), pp. 1–9.Google Scholar
[9]Karaa, S., A hybrid Padé ADI scheme of higher-order for convection-diffusion problems, Int. J. Numer. Meth. Fluids, 64 (2010), pp. 532–548.Google Scholar
[10]LeVeque, R. J., Finite Difference Methods for Ordinary and Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, USA, 2007.Google Scholar
[11]Li, M., Tang, T. and Fornberg, B., A compact fourth-order finite difference scheme for the steady incompressibe Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 20 (1995), pp. 1137–1151.Google Scholar
[12]Li, M. and Tang, T., A compact fourth-order finite difference scheme for unsteady viscous incompressible flows, J. Sci. Comput., 16 (2001), pp. 29–45.Google Scholar
[13]Mahesh, K., A family of high order finite difference schemes with good spectral resolution, J. Comput. Phys., 145 (1998), pp. 332–358.CrossRefGoogle Scholar
[14]Morton, K. W., Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London, UK, 1996.Google Scholar
[15]O’Riordan, E. and Stynes, M., Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems, Adv. Comput. Math., 30 (2009), pp. 101–121.CrossRefGoogle Scholar
[16]Radhakrishna Pillai, A. C., Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, Int. J. Numer. Meth. Fluids, 37 (2001), pp. 87–106.CrossRefGoogle Scholar
[17]Roos, H.-G., Stynes, M. and Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations, Springer, New York, 1996.CrossRefGoogle Scholar
[18]Sanyasiraju, Y. V. S. S. and Mishra, N., Spectral resolutioned exponential compact higher order scheme (SRECHOS) for convection-diffusion equations, Comput. Methods Appl. Mech. Eng., 197 (2008), pp. 4737–4744.CrossRefGoogle Scholar
[19]Sanyasiraju, Y. V. S. S. and Mishra, N., Exponential compact higher order scheme for nonlinear steady convection-diffusion equations, Commun. Comput. Phys., 9 (2011), pp. 897–916.CrossRefGoogle Scholar
[20]Spotz, W. F., High-Order Compact Finite Difference Schemes for Computational Mechanics, Ph.D. Dissertation, the University of Texas at Austin, December 1995.Google Scholar
[21]Spotz, W. F., Accuracy and performance of numerical wall boundary conditions for steady, 2D, incompressible streamfunction vorticity, Int. J. Numer. Meth. Fluids, 28 (1998), pp. 737–757.Google Scholar
[22]Spotz, W. F. and Carey, G. F., High-order compact scheme for the stream-function vorticity equations, Int. J. Numer. Meth. Eng., 38 (1995), pp. 3497–3512.Google Scholar
[23]Stynes, M., Steady-state convection-diffusion problems, Acta Numer., (2005), pp. 445–508.CrossRefGoogle Scholar
[24]Tian, Z. F. and Dai, S. Q., High-order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys., 220 (2007), pp. 952–974.Google Scholar
[25]Tian, Z. F. and Ge, Y. B., A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, J. Comput. Appl. Math., 198 (2007), pp. 268–286.Google Scholar
[26]Yavneh, I., Analysis of a fourth-order compact scheme for convection-diffusion, J. Comput. Phys., 133 (1997), pp. 361–364.Google Scholar