The 1/6 trick

Increased absolute accuracy may always be purchased by reducing $\Delta x$.Increased accuracy relative to the Nyquist frequency may be purchased at a cost of computer time and analytical clumsiness by adding higher-order terms, say,  

$${ \partial^2 \ \over \partial x^2 } \ \ \ \approx \ \ \ { \d... ...\over 12 }\ { \delta^4 \ \over \delta x^4 } \ +\ \ \hbox{etc.}$$ (37)

As $\Delta x$ tends to zero (37) tends to the basic definitions (30) and (31). Coefficients like the 1/12 in (37) may be determined by the Taylor-series method if great accuracy is desired at small k. Or somewhat different coefficients may be determined by curve-fitting techniques if accuracy is desired over some range of k. In practice (37) is hardly ever used, because there is a less obvious expression that offers much more accuracy at less cost! The idea is indicated by  

$${ \partial^2 \ \over \partial x^2 } \ \ \ \approx \ \ \ { {... ...r 1 \ +\ b \, { \Delta x^2 } { \delta^2 \ \over \delta x^2 } }$$ (38)

where b is an adjustable constant. The accuracy of (38) may be numerically evaluated by substituting from (35) to get  

$$\left( \ { \hat k \, \Delta x \over 2 }\ \right)^2 \eq {\si... ...Delta x\over 2} \over 1\ -\ b\,4\ \sin^2\ {k\,\Delta x\over 2}}$$ (39)

The square root of (39) is plotted in Figure 21 for a value of b = 1/6.

 

sixth
Figure 21 Accuracy of the second-derivative representation equation (38) and (39) (for b=1/6) as a function of spatial wavenumber. The sign of the square root of (39) was chosen to agree with k in the range $- \pi$ to $\pi$ and to be periodic outside the range. (Hale)

Taking b in (38) and (39) to be 1/12, then (38), (39) and (37) would agree to second order in $\Delta x$.The 1/12 comes from series expansion, but the 1/6 fits over a wider range and is a value in common use. Francis Muir has pointed out that the value $1/4 - 1/ \pi^2 \ \approx$ 1/6.726 gives an exact fit at the Nyquist frequency and an accurate fit over all lower frequencies! Few explorationists consider the remaining accuracy deficiency of (38) and (39) to be sufficient to warrant interpolation of field-recorded values. Figure 22 compares hyperbolas for various values of b. Observe in Figure 22 that the longest wavelengths travel at the same speed regardless of b. The time axis in Figure 22 is only 256 points long, whereas in practice it would be a thousand or more. So Figure 22 exaggerates the frequency dispersion attributable in practice to finite differencing the x-axis.

 

sixth5
Figure 22 Hyperbolas for $b\ =\ 0 , 1/12 , 1/6.726 , 1/6 , 1/5$.

Let us be sure it is clear how (38) and (39) are put into use. Take b = 1/6. The simplest prototype equation is the heat-flow equation:  

$${ { \partial \ \over \partial t } } \ q \eq { \partial^2 \... ...a x^2 \over 6 }\ \ { { \delta^2 \ \over \delta x^2 } } } \ \ q$$ (40)

Multiply through the denominator:  

$$\left( 1 \ +\ { \Delta x^2 \over 6 }\ { { \delta^2 \ \over ... ...al \ \over \partial t } \ q \ \ \ \approx \ \ \ \delta_{xx} \ q$$ (41)