Page 10 - Laplace transform numerical inversion

Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 - p 8/18
As could be seen in the introduction, though this algorithm usually performs well, it fails at late
time when inverting the test function ˆ
F (p) = Γ
(p+µ) . Indeed, as can be seen in Figure 4,the inverted
function oscillates around the analytical solution f (t) and it unfortunately becomes negative from
around t = 3.3. Indeed, at this instant the sign test indicator becomes negative. One reminds that
an appropriate error behaviour would have been to remain close to zero with a sign test indicator
always being strictly positive. 2 3 4 5 6
-0.5
0 1
0
0.5
1
1.5
f(t)
0 1 2 3 4 5 6
-0.00015
-0.0001
-5e-005
0
5e-005
0.0001
0.00015
Error: f(t) - Fp_inv(t)
Test: Fp_inv(t) < 0
Figure 4: Inversion of f (t) = e− µt by Stehfest N = 14 algorithm.
The precision of the Gaver-Stehfest inversion method depends on the Stehfest number N . Indeed,
one can see in equation(18) that the inversion is based on a summation of N weighted values.
Theoretically, the bigger N , the more precise the inversion (but the slower). Results shown in
Figure 4 were obtained by choosing N = 14. In practice since speed is also important the default
Stehfest number is often chosen around N = 8. The artefact in therefore more important.
At this point, one can assume that, to get closer to the analytical solution, one only needs to use
a bigger Stehfest number N . This actually is what is recommended in the literature. Therefore,
several inversion attempts are tested with bigger Stehfest number N :
• One can see in Figure 5 the result obtained with N = 16 .
å The reconstruction error is smooth and, as expected, smaller than with N = 14.
• In Figure 6 ,one sets N = 18 .
å The reconstruction error is smaller than with N = 16, but it becomes "noisy". This means
that the reconstruction error is now due to two different phenomena:
1. algorithm reconstruction error,
2. numerical approximation error.
• In Figure 7 ,one sets N = 20 .
å The numerical approximation noise becomes predominant, reconstruction error is now
bigger than with N = 18 !
• Finally, in Figure 8,one sets N = 22.
å One can see that the numerical approximation error grows dramatically and highly impacts
the reconstruction result.