5.1.1 FTCS Scheme; Stability Analysis

Writing $\mbox{$\bf u$}_{j}^{n} \equiv \mbox{$\bf u$}(x_{j},t_{n})$ and using DNGF for the time derivative (FT, ``forward-time''), and DST for the space derivative (CS, for ``centered-space''), we write $\partial \mbox{$\bf u$}/\partial t = - \partial \mbox{$\bf j$}/\partial x$ as

$$\frac{1}{\Delta t} \left[ \mbox{$\bf u$}_{j}^{n+1}-\mbox{$\bf u$}_{j}^{n} \right] \approx - \frac{1}{2\, \Delta x} \left[ \mbox{$\bf j$}_{j+1}^{n}-\mbox{$\bf j$}_{j-1}^{n} \right]$$

$$\fbox{$ \displaystyle \mbox{$\bf u$}_{j}^{n+1}= \mbox{$\bf u$}_{j... ... \Delta x} \left[ \mbox{$\bf j$}_{j+1}^{n}-\mbox{$\bf j$}_{j-1}^{n} \right] $ }$$

Stability analysis (J. v. Neumann):

At time $t_{n}$, expand $u(x,t)$:

$$u_{j}^{n} = \sum_{k} U_{k}^{n} e^{ikx_{j}}$$

where $k=2\pi l/L \;\; (l=0,1,\dots)$. Insert this in $u_{j}^{n+1}=T[u_{j'}^{n}]$ to find each Fourier component's propagation law, $U_{k}^{n+1}=g(k) \, U_{k}^{n}$.

$\Longrightarrow$Stable if $\vert g(k) \vert \leq 1 \;\;{\rm for \; all \; k}$.



Application to FTCS + advective equation with $j=c\,u$:

$$g(k)\, U_{k}^{n} e^{ikj \, \Delta x} = U_{k}^{n} e^{ikj \, \Delta x} - \frac{c \, \Delta t}{2\, \Delta x} U_{k}^{n} \, [e^{ik(j+1)\Delta x} - e^{ik(j-1)\Delta x}]$$

or

$$g(k) = 1- \frac{ic \, \Delta t}{\Delta x} \sin \, k \Delta x$$

Obviously, $\vert g(k)\vert > 1$ for any $k$; the FTCS method is inherently unstable.