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# The Convection-Diffusion Equation

Consider the unsteady-state convection-diffusion problem described by the equation:
,
where and are the diffusion coefficient and the velocity, respectively.
The initial and boundary conditions are:
, ,
, ,
, ,
where is the concentration and is the position.
This problem has an analytical solution:
.
This Demonstration plots the solution . The numerical solution obtained using Chebyshev orthogonal collocation is given by the red dots. The analytical solution is given by the blue curve. Excellent agreement between the two solutions is observed. You can vary the values of , , and as well as the number of Chebyshev collocation points, .

### DETAILS

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extrema of the Chebyshev polynomials of the first kind, .
The Chebyshev derivative matrix at the quadrature points is an matrix given by
, , for , and for , , and ,
where for and .
The matrix is then used as follows: and , where is a vector formed by evaluating at , , and and are the approximations of and at the .
References
[1] P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.
[2] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.

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