Examples:

It seems like I run into problems all the time that are good candidates for numerical modeling. Here are a few that I offer to give you a sense of possibilities for your project

Dynamics:  

In almost every problem we did last year we explicitly ignored friction and/or air drag. All such problems make good first numerical problems. Questions like: How does air drag change the range of a hit baseball? Is this effect significantly different in Denver relative to San Francisco? How does air drag on a rotating ball lead to the hooking and bending of football (soccer) kicks?

Another set of dynamics problems that are fun to approach numerically are space based trajectory and orbital mechanics problems. How does a satellite slingshot around a planet and gain speed? Can you figure out the insertion angle for a desired exit direction? What is the orbital path from earth to mars that balances fuel load/consumption with time of travel?

Kinematics:  

A set of kinematics problems that are fun to approach numerically are time of travel calculations in environments where the speed of the object depends on the environment. These include travel times for tsunami where the speed depends on the depth of the ocean. Seismic waves travel through a layered earth and change direction and speed depending on the density of the interior of the earth. This includes changes in direction as well.

 

Electrostatics/dynamics:  

The simulations I showed in class from Falstad are numerical models of the fields produced by various charge distributions. Calculating the charge distribution on an irregular surface and the fields around it is an interesting numerical calculation. Finding the paths of charged particles though the non-uniform magnetic field of the earth and another example.

Thermal:  

There are all sorts of interesting numerical problems in heat transport. These range from dynamic heat flow through thick walls (thermal flywheel effect) to heat flow through complex structures with many materials or changes in profile. Heat flow along turbine blades as they move though the air is a very complex problem. The temperature profile surounding a point source of heat like a laser or a welding arc is an important problem for understanding the impact of heating on the material properties of the surrounding metal (usually).

Differential Equations (non linear oscillators):  

Many problems in Diff Eq are periodic in nature but have complex non-linear terms in them that make it hard to get an analytic solution. Numerical methods can be very helpful in constructing possible solutions to these problems. Many of the problems in Environmental Science and climate change have these sorts of features and make very interesting problems. If these sorts of problems interest you consider looking into 'Consider a Spherical Cow' and Consider a 'Cylindrical Cow' by John Harte.

Optics:  

We won't get to this for a while in PH213 but tracing the path of light through a multilens system or lenses with complex aspheric surfaces is most easily addressed with numerical tools This is how we design the lenses for cell phones these days.