Physics 281: Computational Physics, Spring 2019
Activities for Meetings 16-17, Tues-Thurs 3/26-28/191
• Newman Ch. 8: Ordinary differential equations:
– 8.1 First-order differential equations with one variable.
– 8.2 Differential equations with more than one variable.
– 8.3 Second-order differential equations.
These three sections cover the material we will use in P281. You should
take a look at the remaining sections of Ch. 8 to get an idea of some
more advanced methods for solving ODEs.
• Section “Solving ordinary differential equations” in handout Summary
of numerical methods.
Completing all of the non-starred activities will count as A-level work. To
receive credit, these activities must be checked off in class or office hours
within three weeks (by Fri 4/19).
- Solving an ODE with odeint: The low-pass filter. The circuit
shown in the figure below built from a resistor R and a capacitor C is
called a “low-pass filter”. The input voltage Vin(t) is specified. Then
the output voltage Vout(t) is determined by the following first-order
This is called a low-pass filter because low-frequency sine waves with
angular frequencies ω 1/RC pass through this circuit nearly unchanged, while high frequency sine waves with ω 1/RC are (imperfectly) blocked. For this problem use R = 10 kΩ, C = 1 µF which gives
1/RC = 100 rad/s.
12019-03-27 P281 W9ab.tex
c 2019 Donald Candela
(a) Assume Vin(t) = A sin(ωt) with A = 2.0 V and ω = 50 rad/s.
Write a program that defines a function to compute Vin(t) and
uses this function to make a plot of Vin(t) for 0 < t < tmax, with
tmax chosen to show about two or three complete cycles of the sine
wave. Make the number of time points a variable that you can
easily change it (or input the number of points when you run your
program. Start with several hundred points to get a nice smooth
(b) Write a program to that uses odeint to find Vout(t) at the same
set of time points. Use the initial condition Vout(0) = 0.0 V.
Plot Vin(t) and Vout(t) on the same graph (with labeled axes and
a legend for the two curves, of course). Because ω < 1/RC,
the output should be only a little bit smaller than the input and
slightly shifted in phase.
(c) Run your program again, changing ω to 300 rad/s. Now the output should be quite a bit smaller than the input, and shifted in
phase by nearly π/2. For both values of ω you should find that
Vout(t) is a sine wave except for an initial transient near t = 0.
- Euler’s method. Add code to your program to compute Vout(t) at
the same set of time points using Euler’s method, and plot the results
on the same graph as the odeint results. Find the minimum number of
time points for which the Euler’s-method results appear identical (by
eye) to the odeint results. (For this purpose, you may want to remove
the Vin(t) curve from your plot.)
- RK2. Add code to compute Vout(t) using the second-order RungeKutta method, and plot the RK2, Euler’s-method, and odeint results
together. As with the previous activity, find the minimum number of
time points for which the RK2 results appear identical (by eye) to the
- *RK4. As explained in Newman Sec. 8.1.3, if you didn’t have an
ODE integrator like scipy.integrate.odeint and had to program
one yourself, you would probably use fourth-order Runge-Kutta, which
is quite accurate. Modify your program from the last activity to include
an RK4 integration of the ODE, using the formulas from Newman
Sec. 8.1.3. To check the accuracy, have your program print out y(tmax)
for the various methods (odeint, RK2, RK4).