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. 89
( : 95)



. . >>

u(xj , t ) = where (13.54)
k=’∞

Letting
•k = k∆x,
we have k∆t = »•k and thus

gk = e’ia»•k . (13.55)

The real number •k , here expressed in radians, is called the phase angle of
the k-th harmonic. Comparing (13.54) with (13.52) we can see that γk is
the counterpart of gk which is generated by the speci¬c numerical method
at hand. Moreover, |gk | = 1, whereas |γk | ¤ 1, in order to ensure stability.
Thus, γk is a dissipation coe¬cient; the smaller |γk |, the higher the reduc-
tion of the amplitude ±k , and, as a consequence, the higher the numerical
dissipation.
The ratio a (k) = |γk | is called the ampli¬cation error of the k-th harmonic
k|
|g
associated with the numerical scheme (in our case it coincides with the
ampli¬cation coe¬cient). On the other hand, writing
ω
»•k
’i
γk = |γk |e’iω∆t = |γk |e k ,

and comparing this relation with (13.55), we can identify the velocity of
propagation of the numerical solution, relative to its k-th harmonic, as
being ω . The ratio between this velocity and the velocity a of the exact
k
solution is called the dispersion error d relative to the k-th harmonic
ω ω∆x
d (k) = = .
ka •k a

The ampli¬cation and dispersion errors for the numerical schemes exam-
ined so far are functions of the phase angle •k and the CFL number a».
This is shown in Figure 13.7 where we have only considered the interval
0 ¤ •k ¤ π and we have used degrees instead of radians to denote the
values of •k .
In Figure 13.8 the numerical solutions of equation (13.26) with a = 1
and the initial datum u0 given by a packet of two sinusoidal waves of equal
wavelength l and centred at the origin x = 0 are shown. The plots on the
left-side of the ¬gure refer to the case l = 10∆x while on the right-side
13.9 Dissipation and Dispersion 613

Amplification error for Lax’Friedrichs


1

0.8




a
0.6




µ
CFL= 0.25
0.4 CFL= 0.50
CFL= 0.75
0.2 CFL= 1.00
60
20 80
40 100 120
0 140 160 180
φ

Dispersion error for Lax’Friedrichs


5 CFL= 0.25
CFL= 0.50
CFL= 0.75
4
CFL= 1.00
φ
µ




3

2

1

0 20 40 120
60 100 140 180
80 160
φ

Amplification error for Lax’Wendroff
1.2

1

0.8

0.6
a
µ




CFL= 0.25
0.4
CFL= 0.50
0.2 CFL= 0.75
CFL= 1.00
0
140
120 180
100
80 160
40
0 60
20
φ

Dispersion error for Lax’Wendroff
1.4
1.2
1
0.8
φ
µ




0.6
CFL= 0.25
0.4
CFL= 0.50
CFL= 0.75
0.2
CFL= 1.00
0

120 140 180
160
0 20 100
40 80
60
φ

Amplification error for Upwind
1.2
1
0.8
0.6
a
µ




0.4
CFL= 0.25
CFL= 0.50
0.2
CFL= 0.75
0
CFL= 1.00
’0.2
180
160
140
120
100
0 80
40
20 60
φ

Dispersion error for Upwind
1.4
1.2
1
0.8
µφ




0.6
CFL= 0.25
0.4
CFL= 0.50
CFL= 0.75
0.2
CFL= 1.00
0

140 160 180
120
60 80 100
20 40
0
φ




FIGURE 13.7. Ampli¬cation and dispersion errors for several numerical schemes


we have l = 4∆x. Since k = (2π)/l we get •k = ((2π)/l)∆x, so that
•k = π/10 in the left-side pictures and •k = π/4 in the right-side ones.
All numerical solutions have been computed for a CFL number equal to
0.75, using the schemes introduced above. Notice that the dissipation e¬ect
is quite relevant at high frequencies (•k = π/4), especially for ¬rst-order
methods (such as the upwind and the Lax-Friedrichs methods).
In order to highlight the e¬ects of the dispersion, the same computa-
tions have been repeated for •k = π/3 and di¬erent values of the CFL
number. The numerical solutions after 5 time steps are shown in Figure
13.9. The Lax-Wendro¬ method is the least dissipative for all the consid-
ered CFL numbers. Moreover, a comparison of the positions of the peaks
of the numerical solutions with respect to the corresponding ones in the
614 13. Parabolic and Hyperbolic Initial Boundary Value Problems

Lax’Wendroff CFL= 0.75 φ=π/10
1

0.5

0




u
Computed at t=1
’0.5
Exact
’1
0
’1
’2
’3
’4 2 4
1 3
Lax’Friedrichs CFL= 0.75 φ=π/10
x
1

0.5

0
u




Computed at t=1
’0.5
Exact
’1
4
3
2
1
’1 0
’3 ’2
’4
Upwind CFL= 0.75 φ=π/10
x
1

0.5

0
u




Computed at t=1
’0.5
Exact
’1
4
’4 3
’2 1 2
0
’1
’3
x

Lax’Wendroff CFL= 0.75 φ=π/4
1

0.5
u




0
Computed at t=1
’0.5
Exact
’1
3 4
2
1
0
’1
’3 ’2
’4
Lax’Friedrichs CFL= 0.75 φ=π/4
x
1

0.5

0
u




Computed at t=1
’0.5
Exact
’1
’3
’4 ’2 0 2
’1 1 3 4
Upwind CFL= 0.75 φ=π/4
x
1

0.5

0
u




Computed at t=1
’0.5
Exact
’1
2 3 4
1
0
’1
’3
’4 ’2
x




FIGURE 13.8. Numerical solutions corresponding to the transport of a sinusoidal
wave packet with di¬erent wavelengths


exact solution shows that the Lax-Friedrichs scheme is a¬ected by a posi-
tive dispersion error, since the ”numerical” wave advances faster than the
exact one. Also, the upwind scheme exhibits a slight dispersion error for a
CFL number of 0.75 which is absent for a CFL number of 0.5. The peaks
are well aligned with those of the numerical solution, although they have
been reduced in amplitude due to numerical dissipation. Finally, the Lax-
Wendro¬ method exhibits a small negative dispersion error; the numerical
solution is indeed slightly late with respect to the exact one.


13.9.1 Equivalent Equations
Using Taylor™s expansion to the third order to represent the truncation er-
ror, it is possible to associate with any of the numerical schemes introduced
13.9 Dissipation and Dispersion 615

Lax’Wendroff CFL=0.75 φ=π/3, t=4∆ t
1


0.5




u
0


’0.5
Computed
Exact
’1
’0.5
’1 1
0 2
0.5 1.5


Lax’Wendroff CFL=0.50 φ=π/3, t=4∆ t
1


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. 89
( : 95)



. . >>