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0.5


0
u




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


Lax’Friedrichs CFL=0.75 φ=π/3, t=5∆ t
1


0.5


0
u




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


Lax’Friedrichs CFL=0.50 φ=π/3, t=5∆ t
1


0.5


0
u




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


Upwind CFL=0.75 φ=π/3, t=4∆ t
1


0.5


0
u




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


Upwind CFL=0.50 φ=π/3, t=4∆ t
1


0.5


0
u




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




FIGURE 13.9. Numerical solutions corresponding to the transport of a sinusoidal
wave packet, for di¬erent CFL numbers

so far an equivalent di¬erential equation of the form
vt + avx = µvxx + νvxxx (13.56)
where the terms µvxx and νvxxx represent dissipation and dispersion, re-
spectively. Table 13.2 shows the values of µ and ν for the various methods.
Let us give a proof of this procedure in the case of the upwind scheme.
Let v(x, t) be a smooth function which satis¬es the di¬erence equation
(13.41); then, assuming that a > 0, we have
v(x, t + ∆t) ’ v(x, t) v(x, t) ’ v(x ’ ∆x, t)
+a = 0.
∆t ∆x
Truncating the Taylor expansions of v around (x, t) at the ¬rst and second
order, respectively, we obtain
vt + O(∆t) + avx + O(∆x) = 0 (13.57)
616 13. Parabolic and Hyperbolic Initial Boundary Value Problems

Method µ ν
2
a∆x a ∆t a
’ ’ ∆x2 ’ 3a∆x∆t + 2a2 ∆t2
Upwind
2 2 6
2
a∆x2
∆x
1 ’ (a»)2 1 ’ (a»)2
Lax-Friedrichs
2∆t 3
a∆x2
(a»)2 ’ 1
Lax-Wendro¬ 0
6
TABLE 13.2. Values of dissipation and dispersion coe¬cients for several numer-
ical methods

and
∆t ∆x
vtt + O(∆t2 ) + a vx + vxx + O(∆x2 )
vt + = 0, (13.58)
2 2
where vt = ‚v and vx = ‚x .
‚v
‚t
Di¬erentiating (13.57) with respect to t and then with respect to x, we
get
vtt + avxt = O(∆x + ∆t),
and
vtx + avxx = O(∆x + ∆t).
Thus, it follows that
vtt = a2 vxx + O(∆x + ∆t),
which, after substituting into (13.58), yields the following equation
vt + avx = µvxx (13.59)
where
a∆x a2 ∆t

µ= ,
2 2
and having neglected the term O(∆x2 +∆t2 ). Relation (13.59) is the equiv-
alent di¬erential equation up to second order of the upwind scheme.
Following the same procedure and truncating the Taylor expansion at
third order, yields
vt + avx = µvxx + νvxxx (13.60)
where
a22
a ∆t ’ ∆x2 .
ν=
6
We can give a heuristic explanation of the meaning of the dissipative
and dispersive terms in the equivalent equation (13.56) by studying the
following problem
x ∈ R, t > 0
vt + avx = µvxx + νvxxx
(13.61)
(k ∈ Z)
v(x, 0) = eikx ,
13.9 Dissipation and Dispersion 617

Applying the Fourier transform yields, if µ = ν = 0,



v(x, t) = eik(x’at) , (13.62)



while if µ and ν are arbitrary real numbers (with µ > 0) we get


2 2
v(x, t) = e’µk t eik[x’(a+νk )t]
. (13.63)



Comparing (13.62) with (13.63) we can see that the module of the solution
diminishes as µ grows and this becomes more relevant if the frequency k
gets larger. Therefore, the term µvxx in (13.61) has a dissipative e¬ect on
the solution. A further comparison between (13.62) and (13.63) shows that
the presence of the term ν modi¬es the velocity of the propagation of the
solution; the velocity is increased if ν > 0 whereas it is diminuished if ν < 0.
Even in this case the e¬ect is ampli¬ed at high frequencies. Therefore, the
third-order di¬erential term νvxxx introduces a dispersive e¬ect.
Generally speaking, even-order derivatives in the equivalent equation rep-
resent di¬usive terms, while odd-order derivatives mean dispersive e¬ects.
In the case of ¬rst-order schemes (like the upwind method) the dispersive
e¬ect is often only slightly visible since it is hidden by the dissipative one.
Actually, taking ∆t and ∆x of the same order, we have that ν << µ as
∆x ’ 0, since ν = O(∆x2 ) and µ = O(∆x). In particular, for a CFL
number of 1 , the equivalent equation of the upwind method exhibits null
2
dispersion, truncated at second order, according to the results of the pre-
vious section.
On the other hand, the dispersive e¬ect is strikingly visible in the Lax-
Friedrichs and in the Lax-Wendro¬ schemes; the latter, being second-order
accurate, does not exhibit a dissipative term of the form µvxx . However,
it ought to be dissipative in order to be stable; actually, the equivalent
equation (truncated at fourth order) for the Lax-Wendro¬ scheme reads



a∆x2 a∆x3
[(a») ’ 1]vxxx ’ a»[1 ’ (a»)2 ]vxxxx
2
vt + avx =
6 6


where the last term is dissipative if |a»| < 1. We thus recover the CFL
condition for the Lax-Wendro¬ method.
618 13. Parabolic and Hyperbolic Initial Boundary Value Problems

13.10 Finite Element Approximation of Hyperbolic
Equations
Let us consider the following ¬rst-order linear, scalar hyperbolic problem
in the interval „¦ = (±, β) ‚ R
±
 ‚u + a ‚u + a0 u = f in QT = „¦ — (0, T )

 ‚t

 ‚x

(13.64)
t ∈ (0, T )
 u(±, t) = •(t)





x ∈ „¦,
u(x, 0) = u0 (x)

where a = a(x), a0 = a0 (x, t), f = f (x, t), • = •(t) and u0 = u0 (x) are
given functions.
We assume that a(x) > 0 ∀x ∈ [±, β]. In particular, this implies that
the point x = ± is the in¬‚ow boundary, and the boundary value has to be
speci¬ed there.


13.10.1 Space Discretization with Continuous and
Discontinuous Finite Elements
A semi-discrete approximation of problem (13.64) can be carried out by
means of the Galerkin method (see Section 12.4). De¬ne the spaces

Vh = Xh = vh ∈ C 0 („¦) : vh|Ij ∈ Pr (Ij ), ∀ Ij ∈ Th
r


and
Vh = {vh ∈ Vh : vh (±) = 0} ,
in


where Th is a partition of „¦ (see Section 12.4.5) into n ≥ 2 subintervals
Ij = [xj , xj+1 ], for j = 0, . . . , n ’ 1.
Let u0,h be a suitable ¬nite element approximation of u0 and consider
the problem: for any t ∈ (0, T ) ¬nd uh (t) ∈ Vh such that
± β β


 ‚uh (t) ‚uh (t)

 vh dx + a + a0 (t)uh (t) vh dx

 ‚t ‚x

 ± ±
β (13.65)


 ∀ vh ∈ in
 = f (t)vh dx Vh




 ±
uh (t) = •h (t) at x = ±,

with uh (0) = u0,h ∈ Vh .
13.10 Finite Element Approximation of Hyperbolic Equations 619

If • is equal to zero, uh (t) ∈ Vh , and we are allowed to taking vh = uh (t)
in

and get the following inequality
t t
2
d„ + a(β) u2 („ ) d„
2
uh (t) + µ0 uh („ ) L2 (±,β)
L2 (±,β) h
0 0
t
1
¤ 2
2
u0,h + f („ ) L2 (±,β) d„ ,
L2 (±,β)
µ0
0

for any t ∈ [0, T ], where we have assumed that
1
0 < µ0 ¤ a0 (x, t) ’ a (x). (13.66)
2
Notice that in the special case in which both f and a0 are identically zero,
we obtain
uh (t) L2 (±,β) ¤ u0,h L2 (±,β)
which expresses the conservation of the energy of the system. When (13.66)
does not hold (for example, if a is a constant convective term and a0 = 0),
then an application of Gronwall™s lemma 11.1 yields
t

+ a(β) u2 („ ) d„
2
uh (t) L2 (±,β) h

« 
0 (13.67)
t t

¤  u0,h d„  exp [1 + 2µ— („ )] d„,
2
2
+ f („ ) L2 (±,β)
L2 (±,β)
0 0

where µ— (t) = max|µ(x, t)|.
[±,β]
An alternative approach to the semi-discrete approximation of problem
(13.64) is based on the use of discontinuous ¬nite elements. This choice
is motivated by the fact that, as previously pointed out, the solutions of
hyperbolic problems (even in the linear case) may exhibit discontinuities.
The ¬nite element space can be de¬ned as follows
Wh = Yh = vh ∈ L2 (±, β) : vh|Ij ∈ Pr (Ij ), ∀ Ij ∈ Th ,
r


i.e., the space of piecewise polynomials of degree less than or equal to r,
which are not necessarily continuous at the ¬nite element nodes.
Then, the Galerkin discontinuous ¬nite element space discretization reads:
for any t ∈ (0, T ) ¬nd uh (t) ∈ Wh such that
±x
±β
n’1  i+1


 ‚uh (t) ‚uh (t)

 vh dx + a + a0 (x)uh (t) vh dx
 
 ‚t ‚x
i=0
± xi
(13.68)
 β


 +a(u+ ’ U ’ )(x , t)v + (x ) = f (t)v dx ∀v ∈ W ,


 i i h h h
h h h
±
620 13. Parabolic and Hyperbolic Initial Boundary Value Problems

where {xi } are the nodes of Th with x0 = ±, and for each node xi , vh (xi ) +

denotes the right-value of vh at xi while vh (xi ) is its left-value. Finally,
Uh (xi , t) = u’ (xi , t) if i = 1, . . . , n ’ 1, while Uh (x0 , t) = •(t) ∀t > 0.
’ ’
h
If a is positive, xj is the in¬‚ow boundary of Ij for every j and we set

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. 90
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