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A diffusion equation is of the form

[tex]\frac{\partial u}{\partial t}=k \frac{\partial^2u}{\partial x^2}[/tex]

Usually an equation like this seems to be solved numerically using the Crank-Nicolson method:

[tex]\frac{u_{i}^{n + 1} - u_{i}^{n}}{\Delta t} = \frac{k}{2 (\Delta x)^2}\left((u_{i + 1}^{n + 1} - 2 u_{i}^{n + 1} + u_{i - 1}^{n + 1}) + (u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n})\right)[/tex]

i.e the time-related part is approximated using a 2-point forward difference approximation and the space-part using an average of two 3-point central difference approximation. ([tex]u_{i}^{n}[/tex] means value of u at point i at timestep n, i.e u(i,n))

Is there a specific reason why a central difference is not used for the time? Since a central difference results in a smaller error, yet it's not usually used, what drawback does it have?

What about the central difference for the space-related part? Is there a reason for using the 3-point version instead of the 5-point version? Or is the simpler one usually used because it results in a tridiagonal matrix which is easier to solve?

edit: Clarified notation

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# Approximations used in Crank-Nicolson method for solving PDEs numerically

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