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5.2.1 FTCS Scheme for Parabolic DE

In $\partial u /\partial t = \lambda \partial^{2} u/\partial x^{2}$, replace $\partial u/\partial t$ by DNGF and $\partial^{2} u/\partial x^{2}$ by DDST formula:
$\Longrightarrow$ ``forward time-centered space'' algorithm,
$\displaystyle \frac{1}{\Delta t} \left[ u_{j}^{n+1}-u_{j}^{n}\right]$ $\textstyle =$ $\displaystyle \frac{\lambda}{(\Delta x)^{2}} \left[u_{j+1}^{n}-2 u_{j}^{n}+u_{j-1}^{n}\right]$  

Using $a \equiv \lambda \Delta t / (\Delta x)^{2}$ this is
    $\displaystyle \fbox{$ \displaystyle
u_{j}^{n+1}=(1-2a)u_{j}^{n} + a(u_{j-1}^{n}+u_{j+1}^{n})
$}$  



\begin{figure}\includegraphics[width=120pt]{figures/f5ftcsp.ps}
\end{figure}
Stability:

For the $k$-dependent growth factor we find $
g(k) = 1-4a\,\sin^{2}\frac{k \Delta x}{2}
$, which tells us that for stability the condition is
$\displaystyle \Delta t$ $\textstyle \leq$ $\displaystyle \frac{(\Delta x)^{2}}{2\lambda} \equiv \tau$  

where $\tau$ is the characteristic time for the diffusion over a distance $\Delta x$ (i.e. one lattice space).

The FTCS scheme is simple and stable, but inefficient.





EXERCISE: Remember the thermal conduction problem we considered earlier? If you haven't done it then, do it now, using FTCS. Interpret the behavior of the solution for varying time step sizes in the light of the above stability considerations.



next up previous
Next: 5.2.2 Implicit Scheme of Up: 5.2 Initial Value Problems Previous: 5.2 Initial Value Problems
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001