rinitial condition
8.1 One-Dimensional Flow
411
r-....· _.. _ ......._ - - - - - _._-- - ,
(b)
500 steps
I
50 ) stcps
(c)
I
I
I
:
" l t l l l l i
111 .. ,, 1
FIGURE 8.3. Solution to (8.1) at (a) the initial time, and after (b) 530 and (e) 531 time
steps.
This condition is not satisfied by the previously troublesome mode (w, k)
(7T /!i.t, 0). The only resolvable modes that simultaneously satisfy (8.24) and the
dispersion relation (8.22) are (w , k) = (0, 0) and (7T / !i.t, 7T/ !i.x ), both of which
have outward-directed group velocities. The numericallBVP is therefore stable.
The numerical solution may be altematively stabilized without modifying the
original boundary condition epö = ep' l if the leapfrog time difference is replaced
by the trapezoidal scheme:
epn.+ 1 _ epn. + !.: (epn.+ 1 - epn.+ 1 + ep'!
J
J
4
J+I
J-I
J+I
- ep' !
J-I
) = 0 .
The dispersion relation and group velocity for the preceding trapezoidal scheme
are
tan ( -2W!i.t ) =
J.L
'2 sin k!i.x and aW ak = U cos(k!i.x) cos
2 (W!i.t) -2- .