INTRODUCTION TO STABILITY

 

Experience shows that as soon as you undertake an application that departs significantly from textbook situations, stability becomes a greater concern than accuracy. Stability, or its absence, determines whether the goal is achievable at all, whereas accuracy merely determines the price of achieving it. Here we will look at the stability of the heat-flow equation with real and with imaginary heat conductivity. Since the latter case corresponds to seismic migration, these two cases provide a useful background for stability analysis.

Most stability analysis is based on Fourier transformation. More simply, single sinusoidal or complex exponential trial solutions are examined. If a method becomes unstable for any frequency, then it will be unstable for any realistic case, because realistic functions are just combinations of all frequencies. Begin with the sinusoidal function  

$$P(x) \eq P_0 \ e^{{i\,k\,x}}$$ (62)

The second derivative is  

$${ \partial^2 P \over \partial x^2 } \eq - \, k^2 \, P$$ (63)

An expression analogous to the second difference operator defines $\hat k$:

$$\begin{eqnarray} { \delta^2 P \over \delta x^2 }\ \ \ & =&\ \ \ { P ( x\ +\ \Del... ...-\ \Delta x ) \over \Delta x^2 } \\ \ \ \ &=&\ \ \ - \hat k^2 \, P\end{eqnarray}$$ (64) (65)

Ideally $\hat k$ should equal k. Inserting the complex exponential (62) into (64) gives an expression for $\hat k$:

$$\begin{eqnarray} -\ \hat k^2 \,P \ \ \ &=&\ \ \ { P_0 \over \Delta x^2 } \ \lef... ... k \,\Delta x )^2 \ \ \ &=&\ \ \ 2 \ [ 1\ -\ \cos ( k \Delta x ) ]\end{eqnarray}$$ (66) (67)

It is a straightforward matter to make plots of (67) or its square root. The square root of (67), through the half-angle trig identity, is  

$$\hat k \, \Delta x \eq 2 \ \sin { k \, \Delta x \over 2 }$$ (68)

Series expansion shows that $\hat k$ matches k well at low spatial frequencies. At the Nyquist frequency, defined by $k \Delta x = \pi$,the value of $\hat k \Delta x = 2$ is a poor approximation to $\pi$.As with any Fourier transform on the discrete domain, $\hat k$ is a periodic function of k above the Nyquist frequency. Although k ranges from minus infinity to plus infinity, $\hat k^2$ is compressed into the range zero to four. The limits to the range are important since instability often starts at one end of the range.