A Comparison of Recent Methods for Solving a Model 1D Convection Diffusion Equation

Authors: Ashvin Gopaul, Jayrani Cheeneebash, Kamleshsing Baurhoo

Abstract:

In this paper we study some numerical methods to solve a model one-dimensional convection–diffusion equation. The semi-discretisation of the space variable results into a system of ordinary differential equations and the solution of the latter involves the evaluation of a matrix exponent. Since the calculation of this term is computationally expensive, we study some methods based on Krylov subspace and on Restrictive Taylor series approximation respectively. We also consider the Chebyshev Pseudospectral collocation method to do the spatial discretisation and we present the numerical solution obtained by these methods.

Keywords: Chebyshev Pseudospectral collocation method, convection-diffusion equation, restrictive Taylor approximation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070207

References:

[1] F.S.V. Bazan, "Chebyshev pseudospectral method for computing numerical solution of convection-diffusion equation", Applied Mathematics and Computation, 200, 2008, 537-546.
[2] Chi-Tsong Chen, Linear System Theory and Design, third ed., Oxford University Press, New York, 1999.
[3] M.K. Jain, Numerical solution of differential equations, Wiley Eastern Limited, 1991.
[4] H.N.A. Ismail & E.M.E. Elbarbary, "Restrictive Taylor-s approximation and parabolic partial differential equations", International Journal of Computer Mathematics, 78, 2001, 73-82.
[5] H. N. A. Ismail, E.M. E. Elbarbary, & G.S.E. Salem, "Restrictive Taylor's approximation for solving convection-diffusion equation", Applied Mathematics and Computation, 147, 2004, 355-363.
[6] D. K. Salkuyeh, "On the finite difference approximation to the convection-diffusion equation", Applied Mathematics and Computation, 179, 2006, 79-86
[7] G.D. Smith, Numerical solution of partial differential equations (finite difference methods), Oxford University Press, Oxford, 1990.