50 Computational Modelling in Hydraulic and Coastal Engineering
and the numerical scheme is always unstable. By applying the same procedure to the Equation 3.36, the amplification factor is
G
c t
x
k x
c t
x
k
n
o
o
= −
−
− −
1
1
1
∆
∆
∆
∆
∆
[ cos(
)]
s
π
i in(
)
k x
π∆
(3.41)
and its modulus reads
G
c t
x
c t
x
k
n
o
o
2
1 2
1
1
= −
−
−
∆
∆
∆
∆
[ cos(k k x
π∆ )]
(3.42)
Therefore, the numerical scheme given by Equation 3.36 is stable when
the RHS of Equation 3.42 is less than one, that is, when the CFL criterion
(Equation 3.29) is satisfied.
3.3.3.3 Lax numerical scheme
To avoid instability induced by the second-order centred FD approximation
of the space derivative, a number of numerical schemes have introduced an
artificial controllable numerical diffusion. One of those methods is the Lax
scheme, which is written in the following form:
f
f
t
c
f
f
x
f
f
f
i
n
i
n
o
i
n
i
n
i
n
i
n
i
+
+
−
+
− = −
−
(
) +
−
+
1
1
1
1
2
1
2
2
∆
∆
− −
1
n
t
∆
(3.43)
and leads to the explicit solution
f
f
f
c
t
x
f
f
i
n
i
n
i
n
o
i
n
i
n
+
+
−
+
−
=
+
(
)−
−
(
)
1
1
1
1
1
1
2
2
2
∆
∆
(3.44)
Equation 3.44 introduces an artificial diffusion term, which in combination with the centred FD for the space derivative produces a stable numerical solution as long as the CFL stability criterion is satisfied. Application
of the Lax scheme results in propagation of a signal with the correct speed
but with the formation of trailing oscillations that follow the main signal
(Figure 3.13). That type of numerical error is known as numerical dispersion, and the name is derived from the fact that there is a differential rate
of propagation of signals with different frequencies.
This can be verified using a composite signal that can be analysed to
a number of Fourier (sinusoidal) components. Thus the dispersion would
show through the differentiation of the speed of propagation of the various components with different frequencies. The highest frequency (small
and the numerical scheme is always unstable. By applying the same procedure to the Equation 3.36, the amplification factor is
G
c t
x
k x
c t
x
k
n
o
o
= −
−
− −
1
1
1
∆
∆
∆
∆
∆
[ cos(
)]
s
π
i in(
)
k x
π∆
(3.41)
and its modulus reads
G
c t
x
c t
x
k
n
o
o
2
1 2
1
1
= −
−
−
∆
∆
∆
∆
[ cos(k k x
π∆ )]
(3.42)
Therefore, the numerical scheme given by Equation 3.36 is stable when
the RHS of Equation 3.42 is less than one, that is, when the CFL criterion
(Equation 3.29) is satisfied.
3.3.3.3 Lax numerical scheme
To avoid instability induced by the second-order centred FD approximation
of the space derivative, a number of numerical schemes have introduced an
artificial controllable numerical diffusion. One of those methods is the Lax
scheme, which is written in the following form:
f
f
t
c
f
f
x
f
f
f
i
n
i
n
o
i
n
i
n
i
n
i
n
i
+
+
−
+
− = −
−
(
) +
−
+
1
1
1
1
2
1
2
2
∆
∆
− −
1
n
t
∆
(3.43)
and leads to the explicit solution
f
f
f
c
t
x
f
f
i
n
i
n
i
n
o
i
n
i
n
+
+
−
+
−
=
+
(
)−
−
(
)
1
1
1
1
1
1
2
2
2
∆
∆
(3.44)
Equation 3.44 introduces an artificial diffusion term, which in combination with the centred FD for the space derivative produces a stable numerical solution as long as the CFL stability criterion is satisfied. Application
of the Lax scheme results in propagation of a signal with the correct speed
but with the formation of trailing oscillations that follow the main signal
(Figure 3.13). That type of numerical error is known as numerical dispersion, and the name is derived from the fact that there is a differential rate
of propagation of signals with different frequencies.
This can be verified using a composite signal that can be analysed to
a number of Fourier (sinusoidal) components. Thus the dispersion would
show through the differentiation of the speed of propagation of the various components with different frequencies. The highest frequency (small