2.3 Time-Differencing
51
'!fr(nl:it) = '!frn. The rate at which the solution of a stable finite-difference scheme
converges to the true solution as l:it -+ 0 is one power of l:it lower than the order
of the local truncation error because, roughly speaking, the global error generated
by a stable scheme during an integration over a time interval T is the cumulative
sum of T/ l:it local errors. When the term "truncation error" is used without qualification in this book, it will refer to the global truncation error. The truncation
error of all members of the family of schemes (2.33) is O(l:it), except for the
trapezoidal method, which is 0 [(M)2].
Application of (2.34) to the oscillation equation (2.30) yields
(I - iß« l:it)ifJn+1 = (I + iCXK l:it)ifJn.
The amplification factor for this scheme is
ifJn+1
1+ iaKl:it
1- ißKl:it
.
2 -
-
A:=--=
ifJn
Multiplying A by its complex conjugate gives
1+ a 2K2 M 2
1A 1
---=---=---=- 1+ ß2 K 2l:it2
K 2l:it2
= I + (a
2 - ß2) I + ß2 K2
l:it 2 .
(2.35)
Inspection of (2.35) shows that the scheme is neutral when a = ß, damping
when a < ß, and amplifying when a > ß. The amplification produced when
a > ß is clearly unstable in the sense that approximate solutions computed with
finite Ar can generate floating-point overflows on digital computers, whereas the
magnitude of the correct solution is bounded by lifJül. Note, however, that the
amplification factor for forward differencing satisfies the general Von Neumann
stability condition (2.23), since for l:it
lAIforward = 1 + (Kl:it)2 s
I,
s 1 +K
2l:it .
to yield
lAIforward s
= 1 + (Kl:it)2 s 1 +K
2l:it
As a consequence, the amplifying solutions obtained using forward differencing
do converge to the correct solution of the oscillation equation as l:it -+ O. Forward
differencing also generates convergent approximations to most other ordinary differential equations, but convergence is not guaranteed, and (2.23) is not satisfied,
when forward time differencing is used in conjunction with centered-difference
approximations to the spatial derivative in the advection equation. This point is
discussed further in Section 2.5.
The amplitude and phase errors in the approximate solution are functions of the
numerical resolution. The solution to the goveming differential equation (2.30)
oscillates with aperiod T = 2rr/K. An appropriate measure ofnumerical resolution is the number of time steps per oscillation period, T / l:it. The numerical resolution is improved by decreasing the step size, In the limit of very good numerical
resolution, T / l:it -+ 00 and Kl:it -+ O. Assuming good numerical resolution,
51
'!fr(nl:it) = '!frn. The rate at which the solution of a stable finite-difference scheme
converges to the true solution as l:it -+ 0 is one power of l:it lower than the order
of the local truncation error because, roughly speaking, the global error generated
by a stable scheme during an integration over a time interval T is the cumulative
sum of T/ l:it local errors. When the term "truncation error" is used without qualification in this book, it will refer to the global truncation error. The truncation
error of all members of the family of schemes (2.33) is O(l:it), except for the
trapezoidal method, which is 0 [(M)2].
Application of (2.34) to the oscillation equation (2.30) yields
(I - iß« l:it)ifJn+1 = (I + iCXK l:it)ifJn.
The amplification factor for this scheme is
ifJn+1
1+ iaKl:it
1- ißKl:it
.
2 -
-
A:=--=
ifJn
Multiplying A by its complex conjugate gives
1+ a 2K2 M 2
1A 1
---=---=---=- 1+ ß2 K 2l:it2
K 2l:it2
= I + (a
2 - ß2) I + ß2 K2
l:it 2 .
(2.35)
Inspection of (2.35) shows that the scheme is neutral when a = ß, damping
when a < ß, and amplifying when a > ß. The amplification produced when
a > ß is clearly unstable in the sense that approximate solutions computed with
finite Ar can generate floating-point overflows on digital computers, whereas the
magnitude of the correct solution is bounded by lifJül. Note, however, that the
amplification factor for forward differencing satisfies the general Von Neumann
stability condition (2.23), since for l:it
lAIforward = 1 + (Kl:it)2 s
I,
s 1 +K
2l:it .
to yield
lAIforward s
= 1 + (Kl:it)2 s 1 +K
2l:it
As a consequence, the amplifying solutions obtained using forward differencing
do converge to the correct solution of the oscillation equation as l:it -+ O. Forward
differencing also generates convergent approximations to most other ordinary differential equations, but convergence is not guaranteed, and (2.23) is not satisfied,
when forward time differencing is used in conjunction with centered-difference
approximations to the spatial derivative in the advection equation. This point is
discussed further in Section 2.5.
The amplitude and phase errors in the approximate solution are functions of the
numerical resolution. The solution to the goveming differential equation (2.30)
oscillates with aperiod T = 2rr/K. An appropriate measure ofnumerical resolution is the number of time steps per oscillation period, T / l:it. The numerical resolution is improved by decreasing the step size, In the limit of very good numerical
resolution, T / l:it -+ 00 and Kl:it -+ O. Assuming good numerical resolution,