1
Oct. 12, 2006 ATMS 502 – CS 505 – CSE 566 Jewett
Outline for today’s class:
1. Discussion – exam 1
2. Some comments about nesting
a. Error characteristics
• Interpretation of Courant number
• True solution discretized on f.d. grid
b. Update frequency
• Error increases as sharp gradients reach boundary
3. Space differencing
a. differential-difference equations
b. 2nd-order centered space
!
du j
dt +cuj+1"uj"1
2#x
$
%
&
'
(
) =0; uj(t)=ei kj#x"
*
2ct
( )
!
"i
#
2cu="c
2$x
eik$x"e"ik$x
( )
u %
#
=csin
&
$x
!
c2c=
"
k=csin
#
k$x; for small k$x, c2c%c1&
#
2
6
'
(
)
*
+
,
!
cg2c=
"#
"
k=
"
"
k
csin
$
%x
&
'
( )
*
+ =ccos
$
( )
(corrected)
c. 4th-order centered-space
d. Phase and group velocities (Durran figs. 2.10, 2.12)
• Phase speed errors
• Group velocity errors
