6.4 Examples
155
Fig. 6.5. Heat flux through the isothermal wall at t = 0.12 (left) and temporal
discretization errors in calculated wall heat flux (right) as a function of the time
step size for various schemes (spatial discretization by CDS, 20 x 20 CV uniform
grid)
in Fig. 6.5. As in the previous test case, the results obtained using second
order schemes change little as the number of time steps is increased, while
the first order schemes are far less accurate. The explicit Euler scheme does
not converge monotonically; the solution obtained with the largest time step
lies on the opposite side of the time step independent value from the values
obtained using smaller time steps.
For the sake of error estimation, we obtained an accurate reference solution by using very small time step (At = 0.0003, 400 steps to t = 0.12)
with the Crank-Nicolson scheme. The spatial discretization was the same in
all cases, so again the spatial discretization errors cancel out. By subtracting
the heat flux value calculated using different schemes and time steps from
the reference solution, we obtain the estimates of the temporal discretization
error. These are plotted against normalized time step (normalized with the
largest time step) in Fig. 6.5.
Again, the expected asymptotic convergence rates for first and second
order schemes are obtained. However, the lowest error is now obtained with
the second order scheme with three time levels. This is, as in the previous
example, due t o the dominant effect of the initial error, which turns out to
be smaller when the three level scheme is started with implicit Euler method
than in the Crank-Nicolson scheme. With both second order schemes the
errors are much smaller than with first order schemes: the result is more
accurate with second order schemes using the largest time step than with
first order schemes and an eight times smaller time step!
,320 1
- Crank-Nicolson
Implicit 3 Level
Implicit Euler
Explicit Euler
Exact
w
ID
20
50
I00
.I
.2
.5
I
Number of steps
A t
.310
,305
.300 7
h
I
lmplicit Euler
Explicit Euler
\
: \,
\\
-i ,
\
-
.\.
-
\,
-
\ !
-in
-
\*
-
is
- 0 ,/
\
- 0
-
sL/*O
-
' l l l ! L " l l ' t l l l l " l l " l ! ! ! ! ! ! I ! l g
,001 ----- Ideal Slope
1
155
Fig. 6.5. Heat flux through the isothermal wall at t = 0.12 (left) and temporal
discretization errors in calculated wall heat flux (right) as a function of the time
step size for various schemes (spatial discretization by CDS, 20 x 20 CV uniform
grid)
in Fig. 6.5. As in the previous test case, the results obtained using second
order schemes change little as the number of time steps is increased, while
the first order schemes are far less accurate. The explicit Euler scheme does
not converge monotonically; the solution obtained with the largest time step
lies on the opposite side of the time step independent value from the values
obtained using smaller time steps.
For the sake of error estimation, we obtained an accurate reference solution by using very small time step (At = 0.0003, 400 steps to t = 0.12)
with the Crank-Nicolson scheme. The spatial discretization was the same in
all cases, so again the spatial discretization errors cancel out. By subtracting
the heat flux value calculated using different schemes and time steps from
the reference solution, we obtain the estimates of the temporal discretization
error. These are plotted against normalized time step (normalized with the
largest time step) in Fig. 6.5.
Again, the expected asymptotic convergence rates for first and second
order schemes are obtained. However, the lowest error is now obtained with
the second order scheme with three time levels. This is, as in the previous
example, due t o the dominant effect of the initial error, which turns out to
be smaller when the three level scheme is started with implicit Euler method
than in the Crank-Nicolson scheme. With both second order schemes the
errors are much smaller than with first order schemes: the result is more
accurate with second order schemes using the largest time step than with
first order schemes and an eight times smaller time step!
,320 1
- Crank-Nicolson
Implicit 3 Level
Implicit Euler
Explicit Euler
Exact
w
ID
20
50
I00
.I
.2
.5
I
Number of steps
A t
.310
,305
.300 7
h
I
lmplicit Euler
Explicit Euler
\
: \,
\\
-i ,
\
-
.\.
-
\,
-
\ !
-in
-
\*
-
is
- 0 ,/
\
- 0
-
sL/*O
-
' l l l ! L " l l ' t l l l l " l l " l ! ! ! ! ! ! I ! l g
,001 ----- Ideal Slope
1