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I am in the process of collecting material for a class in computational methods. It will include introductions into numerical methods for ordinary differential equations (Runge-Kutta methods, multi-step methods) and parabolic and hyperbolic partial differential equations (finite differences, spectral methods, finite volumes) and how to implement and use them.

As part of that class, I would very much like to show the students an example where they can compare results from a numerical simulation with a (very simple) real world experiment. Unfortunately, I did not really find a suitable candidate.

So I am asking if one of you can think of a very simple experiment that

  • can be conducted in a normal class room (not a lab)
  • needs only ingredients I can buy in a normal supermarket (or drugstore)
  • can be modelled by a relatively simple mathematical equation - think diffusion equation or linear advection equation
  • does not involve anything dangerous like fire (obviously) ;)

Thanks a lot in advance for your help!

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    $\begingroup$ Are thermometers available? I am thinking of heating up something (for instance in a water bath to keep it at 100 degC and then let it cool down. You can place a thermometer in the center, or in several spots and check the heat equation $\endgroup$ – Toulousain Mar 28 '16 at 7:54
  • $\begingroup$ Thermometer would not be a problem, but heating might be. I don't think I am allowed (or would want) to use a gas burner in a lecture theatre. $\endgroup$ – Daniel Mar 28 '16 at 8:38
  • $\begingroup$ Microwave oven? Electric heating plate? Should the experiment be measurable or a change of colour would be enough? $\endgroup$ – Toulousain Mar 28 '16 at 9:37
  • $\begingroup$ It should provide some measurable or detectable output that can roughly be compared to the output of a computational model. There is no need for a perfect or even very good match, qualitative agreement would suffice. $\endgroup$ – Daniel Mar 28 '16 at 10:58
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    $\begingroup$ All I can think is first order ODEs so I am not sure if it serves. For instance, make a solution of an acid (it can be a weak one such as acetic acid) in a beaker. Put some shells (mussels or clams) and stir. With a pH meter you can see how the acid is neutralised and pH increases. You can change the stirring rate to see how the neutralisation changes. The limiting step is disolution of calcium carbonate so the rate of change should be (almost) proportional to the exchange surface $\endgroup$ – Toulousain Mar 28 '16 at 11:53
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Trace the arc of a ping pong ball you are shooting through the room using a slingshot. The equations that describe this are trivial (gravity acts downward, friction acts in the opposite direction of velocity and is proportional to the square of the speed). You can record the arc on a cell phone and I'm sure there is cell phone software that allows you to take apart a video frame by frame so you can determine the position (and initial velocity) using the first couple of frames.

All you need is a ping pong ball, a rubber band (for the slingshot), a tape measure (behind the person shooting the ball), and a cell phone.

(All of this is inspired by having written this article: http://epubs.siam.org/doi/abs/10.1137/100788604 ).

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  • $\begingroup$ That is an interesting idea, particularly because it would directly includes the students in the experiment. I will have to try it out and see if I can get it to work. $\endgroup$ – Daniel Mar 29 '16 at 7:17
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What about some of the classical physics examples shown on myphysicslab? A pendulum or spring is relatively easy to make / obtain, and furthermore you can observe the period, amplitude and chaoticity of the motion without special tools.

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  • $\begingroup$ Also an interesting idea and may be somewhat easier to do than the ping pong ball. I will have to try it out. $\endgroup$ – Daniel Mar 29 '16 at 7:18
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Elastica theory / Euler Bernoulli beam theory is a very good candidate. Equations are relatively simple (especially in the linear case) and there is generally a very good agreement between experiments and mathematical model. Experimental setup is easy (if you work with very slender beams and low loads) and physical parameters to be identified are geometrical (very easy to measure with a caliper) and material Young Modulus (which could be estimated from engineering books, or left as a tunable parameter to be estimated by some preliminary experiment.)

E.g., with three point bending, record the load--deflection curve. Use initial tangent stiffness to estimate Young Modulus, and then compare experimental to analytical and analytical to numerical results. Afterwards you change boundary conditions and solve other cases...

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    $\begingroup$ Also a very interesting idea, particularly since the class is going to be mainly engineering students. They certainly will have come across this before and be familiar with the problem. $\endgroup$ – Daniel Mar 29 '16 at 11:05

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