2.3 Time-Differencing
63
A comparison of this expression with the asymptotic behavior of the exact amplification factor
shows that the local truncation error of the Asselin-filtered leapfrog scheme is
o[(K
In contrast, the local truncation error of the unfiltered leapfrog scheme
(y = 0) is 0 [(K
Thus, Asselin filtering degrades the global truncation error
of the leapfrog scheme from second order to first order.
The preceding derivation was based on the assumption that (AcP n) = A(cP n).
Is this justified? In practice, the initial condition is not time filtered; one simply
defines cP° == cP°. Thus,
Nevertheless, an application of the Asselin time filter to cP n +I gives
(AcP n) = AcP n + y (AcP n-l ) - 2AcPn + A2cPn)
= A (cP n + Y (? - 2cP n + AcP n)) + Y (AcP n-l ) - AcP n- l )
= A(cP n) + Y (AcP n-l ) - AcP n-I ) ,
from which it follows that
(2.55)
In all cases of practical interest , n » 1 and Y « I ; therefore , (2.55) implies that
A may be factored out of the filtering operation with negligible error, and that
(2.53) is indeed equivalent to (2.52).
In the limit of K
« I, the modulus of the amplification factor for the Asselinfiltered leapfrog scheme may be approximated as
IA+IAsselin-LF
IA-IAsselin-LF
I - 2(1 y)
(1 - 2y) +
Y
2-6y +4y
2
Like other first-order schemes, such as forward differencing and the Matsuno
method, the physical mode in the Asselin-filtered leapfrog scheme has an
amplitude error. The behavior of the computational mode is also notable in that lA_I does not approach zero as IK I -+ O.
The asymptotic behavior of the relative phase change in the physical mode is
R+ Asselin-LF
I +2y
1 + 6(1 _ y) (K
2
.