Implicit equations

Recall that the inflation-of-money equation  

$$q_{t+1} \ -\ q_t \eq r \ q_t$$ (73)

is a simple explicit finite differencing of the differential equation $dq / dt \ \approx \ q$.And recall that a better approximation to the differential equation is given by the Crank-Nicolson form  

$$q_{t+1} \ -\ q_t \eq r\ { q_{t+1} \ +\ q_t \over 2 }$$ (74)

that may be rearranged to  

$$\left( 1 \ - \ {r \over 2 }\ \right) q_{t+1} \eq \left( 1 \ + \ {r \over 2 }\ \right) q_t$$ (75)

or  

$${ q_{t+1} \over q_t } \eq { 1\ +\ r / 2 \over 1\ -\ r / 2 }$$ (76)

The amplification factor (76) has magnitude less than unity for all negative r values, even r equal to minus infinity. Recall that the heat-flow equation corresponds to  

$$r \eq - \ { \sigma \, \Delta t \over c }\ k^2$$ (77)

where k is the spatial frequency. Since (76) is good for all negative r, the heat-flow equation, implicitly time-differenced, is good for all spatial frequencies k. The heat-flow equation is stable whether or not the space axis is discretized (then $k \ \rightarrow \ \hat k$) and regardless of the sizes of $\Delta t$ and $\Delta x$.Furthermore, the 15$^\circ$ wave-extrapolation equation is also unconditionally stable. This follows from letting r in (76) be purely imaginary: the amplification factor (76) then takes the form of some complex number 1 + r/2 divided by its complex conjugate. Expressing the complex number in polar form, it becomes clear that such a number has a magnitude exactly equal to unity. Again there is unconditional stability.

At this point it seems right to add a historical footnote. When finite-difference migration was first introduced many objections were raised on the basis that the theoretical assumptions were unfamiliar. Despite these objections finite-difference migration quickly became popular. I think the reason for its popularity was that, compared to other methods of the time, it was a gentle operation on the data. More specifically, since (76) is of exactly unit magnitude, the output has the same $( \omega , k)$-spectrum as the input. There may be a wider lesson to be learned from this experience: any process acting on data should do as little to the data as possible.