2.5 CombinedTime- and Space-Differencing
93
the Taylor series approximations
sinx x - ix3
can be used to obtain
and arcsinx x + ix3
(2.94)
Ifthe time step is chosen to ensure stability, then 1L 2 < I , [c"I < [c], and the decelerating effect of centered spatial differencing dominates the accelerating effects
of leapfrog-time differencing. As suggested by (2.94), in practical computations
the most accurate results are obtained using a time step such that the maximum
value of IILI is slightly less than one.
Now consider the forward-time one-sided space sehe me
i fJ1l . -ifJ1l.
J
!!.t
J + e J J-I = 0,
!!.X
(2.95)
sometimes referred to as the donor-eell scheme . Substitution of (2.89) into (2.95)
gives
e- iwM _ I = IL (e- ik .1.X - I).
(2.96)
It follows that the exact dispersion relation and the exact solution are obtained in
the special case when JJ., = 1. Further analysis is facilitated by separating (2.96)
into its real and imaginary parts
lAIcosWrßt - 1 = lL(cosk!!.x - 1)
(2.97)
and
IA Isin Wr!!.t = IL sin k S» ,
(2.98)
where, W = Wr + iox, and lAI == e
W j M
is the modulus ofthe amplification factor.
Squaring both sides of (2.97) and (2.98) and adding yields
IAI
2 = 1 - 2JJ.,(l - JJ.,)(l - cosk!!.x),
which implies that the donor-cell seheme is stable and damping for 0 :::: JJ., :::: 1,
and that the maximum damping per time step occurs at IL = i . 8
The discrete dispersion relation
(JJ., sin k Sx
)
Wr = -arctan
1
!!.t
1 + lL(cosk!!.x - 1)
8This stability condition is identical to that obtained via the standard Von Neumann stability analysis in Section 2.3.3 .