64
2. Basic Finite-Difference Methods
1.5
1.0
lAI
0.5
0.0
'
o
0.4
0.8
1.2
1.6
KD..t
FIGURE 2.5. Modulus of the amplification factor for the leapfrog-trapezoidal method as a
function of temporal resolution K b.t. The solid and dashed lines represent the physical and
the computational modes, respectively.
Asselin-Robert filtering increases the phase error; doubling it as y increases from
oto !.
The main problem with the Asselin -filtered leapfrog scheme is its first-order
accuracy. There are two altemative techniques that control the leapfrog computational mode without sacrificing second-order accuracy-the leapfrog-trapezoidal
method and the Magazenkov method . The leapfrog-trapezoidal method (Kurihara 1965; Zalesak 1979) is an iterative scheme in which a leapfrog predictor is
followed by a trapezoidal correction step, i.e.,
c/J* = c/Jn -I + 2I:!.tF(c/Jn),
c/Jn+1 = c/Jn + I:!.t (F(c/Jn) + F(c/J*)) .
2
If this scheme is applied to the oscillation equation, the amplitude and relative
phase changes in the physical mode are
IAILF-trap
(Ktit)4
1 - - 4 - '
RLF- trap
(Ktit)2
1 - -1-2-'
where as usual, these approximations hold for small K I:!.t. Leapfrog-trapezoidal
integrations of the oscillation equation will be stable provided that K I:!.t
..fi.
The amplitude error associated with the leapfrog-trapezoidal scheme is plotted as
a function of temporal resolution in Fig. 2.5.
Magazenkov (1980) suggested that the computational mode could be controlled
by altemating each leapfrog step with a second-order Adams-Bashforth step.
Since the Magazenkov method uses different schemes on the odd and even time
steps, the amplification factor differs between the odd and even steps . In order to