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

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...

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Bibliographic Details
Published in:Journal of Applied Mathematics 2013-01, Vol.2013 (2013), p.460-473-647
Main Author: Appadu, A. R.
Format: Article
Language:English
Subjects:
Advection-diffusion equation
Boundary conditions
Boundary value problems
Differential equations, Partial
Diffusion
Dispersions
Dissipation
Errors
Experiments
Finite difference method
Finite element method
Fluid dynamics
Heat transfer
Mathematical analysis
Mathematical models
Mathematical research
Methods
Numerical analysis
Optimization
Partial differential equations
Studies
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Summary: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.
ISSN:1110-757X
1687-0042
DOI:10.1155/2013/734374