Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T07:34:48.370Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Corresponding author. Email: [email protected]
Email: [email protected]
Get access

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
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

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]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