Journal of Applied Mathematics

Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes

A. R. Appadu

Full-text: Open access

Abstract

Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted by h and k, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to the L1 norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advection-diffusion equation at some values of k and h. Two optimisation techniques are then implemented to find the optimal values of k when h=0.02 for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 734374, 14 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807874

Digital Object Identifier
doi:10.1155/2013/734374

Mathematical Reviews number (MathSciNet)
MR3033565

Zentralblatt MATH identifier
1266.65140

Citation

Appadu, A. R. Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes. J. Appl. Math. 2013 (2013), Article ID 734374, 14 pages. doi:10.1155/2013/734374. https://projecteuclid.org/euclid.jam/1394807874


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References

  • N. Kumar, “Unsteady flow against dispersion in finite porous media,” Journal of Hydrology, vol. 63, no. 3-4, pp. 345–358, 1983.
  • J. Isenberg and C. Gutfinger, “Heat transfer to a draining film,” International Journal of Heat and Mass Transfer, vol. 16, no. 2, pp. 505–512, 1973.
  • V. Guvanasen and R. E. Volker, “Numerical solutions for solute transport in unconfined aquifers,” International Journal for Numerical Methods in Fluids, vol. 3, no. 2, pp. 103–123, 1983.
  • M. Dehghan, “On the numerical solution of the one-dimensional convection-diffusion equation,” Mathematical Problems in Engineering, vol. 2005, no. 1, pp. 61–74, 2005.
  • L. L. Takacs, “A two-step scheme for the advection equation with minimized dissipation and dispersion errors,” Monthly Weather Review, vol. 113, no. 6, pp. 1050–1065, 1985.
  • J. E. Fromm, “A method for reducing dispersion in convective difference schemes,” Journal of Computational Physics, vol. 3, no. 2, pp. 176–189, 1968.
  • K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1994.
  • R. J. Babarsky and R. Sharpley, “Expanded stability through higher temporal accuracy for time-centered advection schemes,” Monthly Weather Review, vol. 125, no. 6, pp. 1277–1295, 1997.
  • R. Smith and Y. Tang, “Optimal and near-optimal advection-diffusion finite-difference schemes. V. Error propagation,” Proceedings of the The Royal Society of London A, vol. 457, no. 2008, pp. 803–816, 2001.
  • C. Hirsch, Numerical Computation of Internal and External Flows, vol. 1, John Wiley & Sons, New York, NY, USA, 1988.
  • A. R. Appadu and M. Z. Dauhoo, “The concept of minimized integrated exponential error for low dispersion and low dissipation schemes,” International Journal for Numerical Methods in Fluids, vol. 65, no. 5, pp. 578–601, 2011.
  • A. R. Appadu, “Some applications of the concept of minimized integrated exponential error for low dispersion and low dissipation,” International Journal for Numerical Methods in Fluids, vol. 68, no. 2, pp. 244–268, 2012.
  • A. Mohammadi, M. Manteghian, and A. Mohammadi, “Numerical solution of the one- dimensional advection-diffusion equation using simultaneously temporal and spatial weighted parameters,” Australian Journal of Basic and Applied Sciences, vol. 5, no. 6, pp. 1536–1543, 2011.
  • M. Dehghan, “Weighted finite difference techniques for the one-dimensional advection-diffusion equation,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 307–319, 2004.
  • R. E. Mickens, Applications of Nonstandard Finite Difference Schemes, World Scientific Publishing, River Edge, NJ, USA, 2000.
  • R. E. Mickens, “Analysis of a new finite-difference scheme for the linear advection-diffusion equation,” Journal of Sound and Vibration, vol. 146, no. 2, pp. 342–344, 1991.
  • A. C. Hindmarsh, P. M. Gresho, and D. F. Griffiths, “The stability of explicit euler timeintegration for certain finite difference approximations of the multi-dimensional advection-diffusion equation,” International Journal for Numerical Methods in Fluids, vol. 4, no. 9, pp. 853–897, 1984.
  • E. Sousa, “The controversial stability analysis,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 777–794, 2003.
  • C. K. W. Tam and J. C. Webb, “Dispersion-relation-preserving finite difference schemes for computational acoustics,” Journal of Computational Physics, vol. 107, no. 2, pp. 262–281, 1993.
  • C. Bogey and C. Bailly, “A family of low dispersive and low dissipative explicit schemes for flow and noise computations,” Journal of Computational Physics, vol. 194, no. 1, pp. 194–214, 2004.
  • A. R. Appadu, “Comparison of some optimisation techniques for numerical schemes discretising equations with advection terms,” International Journal of Innovative Computing and Applications, vol. 4, no. 1, pp. 12–27, 2012.
  • C. K. W. Tam and H. Shen, “Direct computation of nonlinear acoustic pulses using high-order finite differences schemes,” AIAA Paper 93-4325, 1993.