Alternating Group Explicit Iterative Methods for One-Dimensional Advection-Diffusion Equation

Abstract

The finite difference method such as alternating group iterative methods is useful in numerical method for evolutionary equations and this is the standard approach taken in this paper. Alternating group explicit (AGE) iterative methods for one-dimensional convection diffusion equations problems are given. The stability and convergence are analyzed by the linear method. Numerical results of the model problem are taken. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show that the behavior of the method with emphasis on treatment of boundary conditions is valuable.

Share and Cite:

Chen, N. and Gu, H. (2015) Alternating Group Explicit Iterative Methods for One-Dimensional Advection-Diffusion Equation. American Journal of Computational Mathematics, 5, 274-282. doi: 10.4236/ajcm.2015.53025.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Celia, M.A., Russell, T.F., Herrera, I. and Ewing, R.E. (1990) An Eulerian-Langrangian Localized Adjoint Method for the Advection-Diffusion Equation. Advances in Water Resources, 13, 187-206. http://dx.doi.org/10.1016/0309-1708(90)90041-2 [2] Dehgan, M. (2004) Weighted Finite Difference Techniques for the One-Dimensional Advection Diffusion Equation. Applied Mathematics and Computation, 147, 307-319. http://dx.doi.org/10.1016/0309-1708(90)90041-2 [3] Evans, D.J. and Sahimi, M.S. (1988) The Alternating Group Explicit (AGE) Iterative Method for Solving Parabolic Equations. I: Two-Dimensional Problems. International Journal of Computer Mathematics, 24, 311-341. http://dx.doi.org/10.1080/00207168808803651 [4] Evans, D.J. and Sahimi, M.S. (1989) The Alternating Group Explicit (AGE) Iterative Method to Solve Parabolic and Hyperbolic Partial Differential Equations. Annual Review of Numerical Fluid Mechanics and Heat Transfer, 2, 283-389. http://dx.doi.org/10.1615/AnnualRevHeatTransfer.v2.100 [5] Evans, D.J. and Sahimi, M.S. (1990) The Solution of Nonlinear Parabolic Partial Differential Equations by the Alternating Group Explicit (AGE) Method. Computer Methods in Applied Mechanics and Engineering, 84, 15-42. http://dx.doi.org/10.1016/0045-7825(90)90087-3 [6] Evans, D.J. and Sahimi, M.S. (1992) The AGE Solution of the Bi-Harmonic Equation for the Deflection of Uniformly Loaded Square Plate. Report 748, Department of Computer Studies, Loughborough University, Loughborough. [7] Sahimi, M.S. and Evans, D.J. (1994) The Numerical Solution of a Coupled System of Elliptic Equations Using the AGE Fractional Scheme. International Journal of Computer Mathematics, 50, 65-87. http://dx.doi.org/10.1080/00207169408804243 [8] Saul. yev V K. (1998) Integration of Techniques for Fluid Dynamics. Springer-Verlag, Berlin. [9] Kellog, G.R.B. (1964) An Alternating Direction Method for Operator Equations. SIAM Journal on Numerical Analysis, 12, 848-854. http://dx.doi.org/10.1137/0112072 [10] Wang, W.Q. (2002) A Class Alternating Segment Method for Solving Convection-Diffusion Equation. Numerical Mathematics, A Journal of Chinese Universities, 4, 289-297.