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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

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

is
 
    $\displaystyle \frac{T(x_0, y_0, t_0 + \Delta t) - T(x_0, y_0, t_0)}
{\Delta t}$  
    $\displaystyle \quad =
\frac{D}{h^2}
\big(
T(x_0 + h, y_0, t_0) + T(x_0 - h, y_0, t_0)$  
    $\displaystyle \qquad + T(x_0, y_0 + h, t_0) + T(x_0, y_0 - h, t_0)$  
    $\displaystyle \qquad - 4 T(x_0, y_0, t_0)
\big)$ (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

 
    $\displaystyle \frac{T(x_0, y_0, t_0 + \Delta t) - T(x_0, y_0, t_0)}
{\Delta t}$  
    $\displaystyle \quad =
\frac{D}{h^2}
\big(
T(x_0 + h, y_0, t_0 + \Delta t) + T(x_0 - h, y_0, t_0 + \Delta t)$  
    $\displaystyle \qquad + T(x_0, y_0 + h, t_0 + \Delta t)
+ T(x_0, y_0 - h, t_0 + \Delta t)$  
    $\displaystyle \qquad - 4 T(x_0, y_0, t_0 + \Delta t)
\big)$ (3.21)

This is called the implicit   integration scheme.


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