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Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes
Author
Source
Journal of Applied Mathematics
Issue
Vol. 2013, Issue 2013 (31 Dec. 2013), pp.1-14, 14 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2013-03-07
Country of Publication
Egypt
No. of Pages
14
Main Subjects
Abstract EN
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.
American Psychological Association (APA)
Appadu, A. R.. 2013. Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes. Journal of Applied Mathematics،Vol. 2013, no. 2013, pp.1-14.
https://search.emarefa.net/detail/BIM-494401
Modern Language Association (MLA)
Appadu, A. R.. Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes. Journal of Applied Mathematics No. 2013 (2013), pp.1-14.
https://search.emarefa.net/detail/BIM-494401
American Medical Association (AMA)
Appadu, A. R.. Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes. Journal of Applied Mathematics. 2013. Vol. 2013, no. 2013, pp.1-14.
https://search.emarefa.net/detail/BIM-494401
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-494401