Commun. Comput. Phys., 9 (2011), pp. 897-916.

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

Y. V. S. S. Sanyasiraju 1*, Nachiketa Mishra 1

1 Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India.

Received 17 December 2009; Accepted (in revised version) 30 July 2010
Available online 13 October 2010


This paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is O(h^4) 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 O(h^4+k^4) 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.

AMS subject classifications: 65N05, 76M20, 35Q35

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Key words: Finite difference, higher order, exponential compact, convection-diffusion.

*Corresponding author.
Email: (Y. V. S. S. Sanyasiraju), (N. Mishra)

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