05 is the difference e n uniformly small throughout

05. Is the difference ∆ E
n
uniformly
small throughout the cycle? Does ∆ E
n
drift, that is, become bigger with time? What is the
optimum choice of ∆ t ?
b. Use the Euler-Cromer algorithm to answer the same questions as in part (a).
c. Modify the program so that the Euler-Richardson algorithm is used and answer the same ques-
tions as in part (a).
d. Compute ∆ E
n
and describe the qualitative difference between the time dependence of ∆ E
n
us-
ing the three algorithms. Which method is most consistent with the requirement of conservation
of energy? For fixed ∆ t , which algorithm yields better results for the position in comparison to
the analytical solution ( 6.4 )? Is the requirement of conservation of energy consistent with the
relative accuracy of the computed positions? For which algorithm does the total energy drift
the least?
e. Choose the best algorithm based on the criteria of your choice and determine the values of ∆ t
that are needed to conserve the total energy to within 0.1% over one cycle for ω
0
= 3 and for
ω
0
= 6. Can you use the same value of ∆ t for both values of ω
0
? If not, how do the values of
∆ t correspond to the relative values of the period in the two cases?
f. It might occur to you that adding a correction for the acceleration to the Euler method would
produce more accurate results. The Verlet algorithm is based on this idea. One form of this
algorithm is given by:
x
n +1
= x
n
+ v
n
∆ t +
1
2
a
n
(∆ t )
2
, (6.9a)
and
v
n +1
= v
n
+
1
2
( a
n +1
+ a
n
)∆ t. (6.9b)
Modify the program so that the Verlet algorithm is used.
You might have noticed that modifying your program each time so that a different algorithm
can be tested is not very object oriented. In Chapter 7 we will introduce the ODE interface to allow
changes in the algorithm to be made easily.
6.3 Threads
In Chapter 5 and Section 6.2 we simulated the motion of a particle by solving Newton’s equations
of motion using several simple numerical methods. The solutions that we obtained were plotted,
but we did not attempt to visualize the motion as it occurred, and all our plots were generated
after the calculations were completed. You probably also noticed that our programs ran for a
number of time steps that was determined by a while statement or predetermined by the user.

CHAPTER 6. OSCILLATIONS AND THE SIMULATION INTERFACE 101
Hence, we could not intervene to stop the simulation or change a parameter. In the following we
will find that in order to visualize the motion as it occurs or to intervene during the simulation to
change a parameter, it is necessary to use threads .
The Calculation interface has an important limitation — it is unable to respond to the click
of a button while the calculation is being performed. This lack of response is not a problem if the
calculation is short. However, if the calculation takes a long time, the user might assume that the
program has crashed or is an infinite loop.