# College Math Teaching

## November 22, 2014

### One upside to a topologist teaching numerical analysis…

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 $y' = -\lambda y$, $y(0) = 1$ for $\lambda > 0$. Example: $y' = -20y, y(0) = 1$. Note that the solution $y(t) = e^{-20t}$ 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: $w_{i+1} = w_{i} + h f(t_i, w_i)$ which becomes $w_{i+1} = w_i -20h \lambda w_i$. There is a way of assigning a method to a polynomial; in this case the polynomial is $p(\mu) = \mu - (1+h\lambda)$ and if the roots of this polynomial have modulus less than 1, then the method will converge. Well here, the root is $(1+h\lambda)$ and calculating: $-1 > 1+ h \lambda > 1$ which implies that $-2 > h \lambda > 0$. This is a good reference.

So for $\lambda = 20$ we find that $h$ has to be less than $\frac{1}{10}$. And so I ran Euler’s method for the initial problem on $[0,1]$ 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.