8.1 One-Dirnensional Flow
407
(al
(bl
(cl
1
I
....
...... 1
F1GURE 8.2. Approximate solution to the difTerential-difTerenee equation (8.14) at time:
(a) 0, (b) 0.25, and (e) 0.75.
After substituting for w from the dispersion relation (8.16), the preceding simplifies to
[r] =1
1 - coskßx 1= tan
1 + coskßx
2 (kßX) .
2
Thus, the magnitude of the reftection coefficient generated by one-sided differencing is identical to that produced by second-order extrapolation.
The interaction of physical and computational modes with the lateral boundaries is illustrated in Fig. 8.2, which is pattemed after an example in Trefethen
(1985). The differential-difference advection equation was solved on the domain
o :s x :s I, with U = I, ßx = 0.0 I, and the initial distribution of rPi was given
by the rectified Gaussian
rPi = 0.5 [I + (_1)i] e-400(.<-0.5)2.
The inftow condition was rPo = 0, and the outftow condition rPN = rPN -I.