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### Solving the Time Dependent Diffusion Problem

Let us discretise our full time dependent Diffusion Equation  both in space and in time. One of the ways to discretise

is

 (3.20)

This is called the explicit   integration scheme.

But this is not the only way. Another way, which seems just as good, but is, in fact, a lot better, is

 (3.21)

This is called the implicit   integration scheme.

• Explicit schemes are very easy to program. In case of the Diffusion Equation they can be all reduced to the   Jacobi scheme.
• Explicit schemes can take a very long time to converge.
• The (Richard) Courant - (Hans) Lewy stability condition very   severely restricts the length of the time step for parabolic problems.
• The Jacobi   scheme corresponds to the longest practical time step that is still compatible with the Courant-Lewy condition.
• The   implicit scheme is unconditionally stable, sic! But extremely hard to program. It results in a very large linear system. For example, for a grid you must solve 40,000 simultaneous linear equations for 40,000 unknowns. The equation matrix alone, if you were so insane as to try to code it literally, would have entries. Luckily the matrix is sparse   and the problem is solvable.
• There is an even better way to tackle the Diffusion Equation, which is a lot easier to program than the implicit scheme, but unlike the explicit scheme, solves the whole problem without iterations in one swoop: the spectral method, i.e., a method that is based on Fourier Transform.

Next: More about the Explicit Up: Fields and Rasters Previous: More about Parallel execution
Zdzislaw Meglicki
2001-02-26