2.3 Time-Differencing
69
Max s
Melhod
Storage
Efficiency Amplification
Phase
Factor
Faclor
Factor
Error
Forward
Backward
2
*
s2
1 - -
0
00
0
00
s2
1- -
s2
1+ 2"
s 2
1- -
2
3
3
Asselin
Leapfrog
3
< 1
y s2
1 - - -2(1 - y)
(I + 2y)s2
1+ 6(1- y)
< 1
Leapfrog
AdamsBashforth-2
Trapezoidal
3
*
2
0
00
s4
1+4"
s2
1+"6
1 + _s2
5
12
s2
1 - -
12
0
00
Runge-Kutta-2
Magazenkov
LeapfrogTrapezoidal
2
3
3
0
0.67
0.71
s4
1+
8
s4
1- -
s4
1 - -
s2
1+"6
s2
1 +"6
s2
1 - -
12
0
0.67
1.41
4
4
AdamsBashforth-3
4
0.72
13 4
- s
8
289 4
1+ -s
720
0.72
AdamsMoulton-3
ABM PredictorCorrector-3
Runge-Kutta-3
4
2
*
0
0.60
0.58
s4
1 + 24
19 4
I - - s
144
s4
I - -
24
11 4
1 - - s
720
1243 4
1 + 8640s
s4
1+ 30
0
1.20
1.73
Runge-Kutta-4
4 t
0.70
s6
1 - -
144
s4
1 - -
120
2.82
t A storage factor of 3 may be achieved following the algorithm of Blum (1962).
TABLE 2.2. Characteristics of the schemes Iisted in Table 2.1. The amplification factor
and relative phase change are for well-resolved solutions to the oscillation equation, and
S = K ßt. "Max s" is the maximum value of K ßt for which the solution is nonamplifying.
The storage and efficiency factors are defined in the text. No storage factor is given for
implicit schemes.