Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T12:12:17.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential Compact Higher Order Scheme for Nonlinear Steady Convection-Diffusion Equations

Published online by Cambridge University Press:  20 August 2015

Y. V. S. S. Sanyasiraju  and
Nachiketa Mishra
Y. V. S. S. Sanyasiraju*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
Nachiketa Mishra*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

This paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.

Keywords

65N0576M2035Q35Finite differencehigher orderexponential compactconvection-diffusion
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 4 , April 2011 , pp. 897 - 916
Copyright
Copyright © Global Science Press Limited 2011

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]Shah, A., Guo, H. and Yuan, L., A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier-Stokes equations, Commun. Comput. Phys., 5 (2009), 712–729.Google Scholar
[2]Allen, D. N. de G. and Southwell, R. V., Relaxation methods applied to determine the motion, in two dimensions, of a viscous fluid past a fixed cylinder, Quart. J. Mech. Appl. Math., 8 (1955), 129–145.Google Scholar
[3]Carpenter, M. H., Gottlieb, D. and Abarbanel, S., Stable and accurate boundary treatments for compact, high-order finite-difference schemes, Appl. Numer. Math., 12 (1993), 55–87.Google Scholar
[4]Gartland, E. C. Jr, Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem, Math. Comput., 48(178) (1987), 551–564.Google Scholar
[5]Il′in, A. M., A difference scheme for a differential equation with a small parameter multiplying the highest derivative (Russian), Math. Zametki., 6 (1969), 237–248.Google Scholar
[6]Kalita, J. C., Dalal, D. C. and Dass, A. K., A fully compact HOC simulation of the steady-state natural convection in a square cavity, Phys. Rev. E., 64(6) (2001), 1–13.Google Scholar
[7]Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103(1) (1992), 16–42.Google Scholar
[8]Li, M., Tang, T. and Fornberg, B., A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations, Int. J. Numer. Methods. Fluids., 20 (1995), 1137–1151.Google Scholar
[9]Mahesh, K., A family of high order finite difference schemes with good spectral resolution, J. Comput. Phys., 145 (1998), 332–358.Google Scholar
[10]Pillai, A. C. R., Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, Int. J. Numer. Methods. Fluids., 37(1) (2001), 87–106.Google Scholar
[11]Roos, H.-G., Stynes, M. and Tobiska, L., Numerical Solution of Convection-Diffusion Problems: Convection-Diffusion and Flow Problems, Springer Series in Computational Mathematics 24, Springer-Verlag, New York, 1996.Google Scholar
[12]Sanyasiraju, Y. V. S. S. and Manjula, V., Higher order semi compact scheme to solve transient incompressible Navier-Stokes equations, Comput. Mech., 35(6) (2005), 441–448.Google Scholar
[13]Sanyasiraju, Y. V. S. S. and Mishra, N., Spectral resolutioned exponential compact higher order scheme (SRECHOS) for convection-diffusion equation, Comput. Methods. Appl. Mech. Eng., 197 (2008), 4737–4744.Google Scholar
[14]Spotz, W. F. and Carey, G. F., High-order compact scheme for the steady stream-function vorticity equations, Int. J. Numer. Methods. Eng., 38 (1995), 3497–3512.CrossRefGoogle Scholar
[15]Tian, Z. F. and Dai, S. Q., High-order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys., 220(2) (2007), 952–974.Google Scholar
[16]You, D., A high-order Padé ADI method for unsteady convection-diffusion equations, J. Comput. Phys., 214(1) (2006), 1–11.Google Scholar