Yes, I was glad when we hired people with applied mathematics expertise; though I am enjoying teaching numerical analysis, it is killing me. My training is in pure mathematics (in particular, topology) and so class preparation is very intense for me.
But I so love being able to show the students the very real benefits that come from the theory.
Here is but one example: right now, I am talking about numerical solutions to “stiff” differential equations; basically, a differential equation is “stiff” if the magnitude of the differential equation is several orders of magnitude larger than the magnitude of the solution.
A typical example is the differential equation , for . Example: . Note that the solution decays very quickly to zero though the differential equation is 20 times larger.
One uses such an equation to test a method to see if it works well for stiff differential equations. One such method is the Euler method: which becomes . There is a way of assigning a method to a polynomial; in this case the polynomial is and if the roots of this polynomial have modulus less than 1, then the method will converge. Well here, the root is and calculating: which implies that . This is a good reference.
So for we find that has to be less than . And so I ran Euler’s method for the initial problem on and showed that the solution diverged wildly for using 9 intervals, oscillated back and forth (with equal magnitudes) for using 10 intervals, and slowly converged for using 11 intervals. It is just plain fun to see the theory in action.