This nondimensionalization is also very nice because there are now no constants in the equation. The nondimensionalization is not that crucial, however, because in this case both components of the solution evolve on basically the same timescale and also this timescale does not change drastically with the motion. You will see as we go on in this course that the nondimensionalization can be much more important, particularly when the model involves two or more timescales. You can now go ahead and solve the problem on page 91. Perhaps the most important choice for you to make when choosing a scheme is the choice of explicit versus implicit.
Explicit schemes run much faster.
However, as we will discuss in class all explicit schemes have a stability limit on the size of the timestep that you can take. When the stable timestep is of the same order as the timescale on which the solution evolves then you can get away with an explicit scheme. On the other hand, when the stable timestep is much smaller than the timescale on which the solution evolves or the timescale that you are interested in then you would want to go to an implicit scheme. We will analyze solution timescales and stable timescales in class. This is generally done using linear problems.