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

$$\frac{\partial T(x, y, t)}{\partial t} = D \nabla^2 T(x, y, t)$$

is

 

$$\frac{T(x_0, y_0, t_0 + \Delta t) - T(x_0, y_0, t_0)} {\Delta t}$$

$$\quad = \frac{D}{h^2} \big( T(x_0 + h, y_0, t_0) + T(x_0 - h, y_0, t_0)$$

$$\qquad + T(x_0, y_0 + h, t_0) + T(x_0, y_0 - h, t_0)$$

$$\qquad - 4 T(x_0, y_0, t_0) \big)$$ (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

 

$$\frac{T(x_0, y_0, t_0 + \Delta t) - T(x_0, y_0, t_0)} {\Delta t}$$

$$\quad = \frac{D}{h^2} \big( T(x_0 + h, y_0, t_0 + \Delta t) + T(x_0 - h, y_0, t_0 + \Delta t)$$

$$\qquad + T(x_0, y_0 + h, t_0 + \Delta t) + T(x_0, y_0 - h, t_0 + \Delta t)$$

$$\qquad - 4 T(x_0, y_0, t_0 + \Delta t) \big)$$ (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 $200\times200$ 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 $40,000\times40,000 = 1,600,000,000$ 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.