One-dimensional diffusion in cylindrical and spherical geometry

There are certain practical diffusion problems, which can be treated most appropriately in cylindrical or in spherical coordinates. In many cases, choosing the natural coordinate system allows for the coordinates to be separated, and one is left with the simpler problem of dealing with one-dimensional diffusion along the radial coordinate. Basically, the only technical complication which arises as compared to the one-dimensional diffusion in Cartesian coordinates treated so far, concerns the approximation of the spatial derivative of the concentration involved by the diffusion equation. 

Assuming constant diffusion coefficient, the equation describing the radial diffusion in cylindrical coordinates may be written  

In order to approximate its solution, we establish a grid of equally spaced points in the interval [0, R] of the radial coordinates. Denoting the corresponding mesh constant by h, the gridpoints are defined by the discrete coordinates  

A second order representation of the right-hand side of Eq. (8-58) at point (r , tn) is obtained by using centered-differences for the spatial derivatives  

A special treatment has to be applied to the central gridpoint n = 0. The singularity arising in the second term is overcome by imposing the natural boundary condition that the first order derivative vanishes at n = 0. By introducing a fictitious gridpoint r0 = ri - h, this condition may be approximated by the second order centered-differ-ence scheme  

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