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