Geoscience Reference
In-Depth Information
Models with
explicit
numerical schemes
use
t
=
t
, while models with
implicit
*
schemes consider
It can be seen from the figure that when the slope of
the curve is positive, explicit models underestimate the advective fluxes,
t
=
t
+ ∆
t.
*
while when
the slope is negative, they overestimate them, introducing (at least) a phase error.
Implicit schemes, on the other hand, underestimate or overestimate the fluxes by a
value of the same order. The consideration of an intermediate value between
†
t
and
t
+ ∆
t
generates more accurate fluxes. The next subsection shows that
t
=
t
+
/
∆
t
*
1
2
(semi-implicit method) gives the maximum accuracy. Values at
t
=
t
+
/
∆
t
can
*
1
2
be obtained by averaging the values of the properties calculated at time
t
and time
t
t.
An increasing number of calculations to perform is the price to pay for
accuracy improvement.
The next subsection shows that implicit methods have better stability properties
than explicit methods. It can be shown that stability properties of the semi-implicit
methods are similar to those of implicit methods. Because of their stability and accu-
racy properties, semi-implicit methods are the most efficient numerical methods.
+ ∆
6.2.4
T
AYLOR
S
ERIES
A
PPROACH
Traditionally, discretized equations are obtained from partial differential equations
by replacing derivatives with finite-differences obtained using the Taylor series. The
Taylor series provides information on the truncation errors arising when replacing
derivatives by finite-differences. In contrast, the control volume introduced in the
previous subsection gives information about physical approaches used during dis-
cretization. When applied correctly, both methods must produce the same discretized
equations.
In order to introduce the Taylor series discretization methods and to analyze
stability and accuracy concepts, let us consider the differential equation correspond-
ing to Equation (6.2):
2
∂
∂
C
t
∂
∂
C
x
∂
∂
C
x
+
U
=
ν
(6.3)
2
This equation describes the advection-diffusion transport in a channel with uniform
velocity, a permanent geometry, and diffusivity.
6.2.4.1
Time Discretization
The Taylor series relates the value of a property in a point (or time instant) with the
values of the property in another point and the derivatives in the same point:
t
t
t
t
22
2
+
33
3
++
nn
n
+
=+
∂
∂
C
t
+
∆
t
∂
∂
C
∆
t
∂
∂
C
t
∆
t
n
∂
∂
C
t
t
+
∆
t
t
n
+
1
CC t
∆
L
0
()
∆
t
i
i
2
t
3
!
!
i
i
i
i
†
In explicit methods the flux during a time step is proportional to the area of the rectangle with side
lengths ∆
t
and
C
t
, while in implicit methods it is proportional to ∆
t
and
C
t
+∆
t
.
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