Basic discretization techniques

What is a derivative?

Limit definition of a derivative:

$ \frac{d u(x)}{dx} = \displaystyle{ \lim_{\Delta x \rightarrow 0} } \frac{ u(x_0 + \Delta x) - u(x_0) }{ \Delta x } $

This is saying, when $ \Delta x $ is "small enough", the algebraic difference $ \frac{ u(x_0 + \Delta x) - u(x_0) }{ \Delta x} $ is a good approximation for the derivative; if $ \Delta x $ is small enough, the representation becomes exact.

OK, so naturally we want to know: how small? How small does $ \Delta x $ have to be?

Using a Taylor series expansion for $ u(x_0 + \Delta x) $ in terms of $ u(x_0) $:

$ u(x_0 + \Delta x) = u(x_0) + \left. \frac{\partial u}{\partial x} \right|_{0} \Delta x + \left. \frac{\partial^2 u}{\partial x^2} \right|_{0} \frac{(\Delta x)^2}{2!} + \dots + \left. \frac{\partial^n u}{\partial x^n} \right|_{\xi} \frac{ (\Delta x)^n }{ n! } \qquad x_0 \leq \xi \leq x_0 + \Delta x $

What does the Taylor series tell us about how small $ \Delta x $ needs to be?

Rearrange the Taylor series to look like the limit definition of the derivative...

$ \left. \frac{\partial u}{\partial x} \right|_{x_0} = \frac{ u(x_0 + \Delta x) - u(x) }{ \Delta x } - \left. \frac{\partial^2 u}{\partial x^2} \right|_0 \frac{\Delta x}{2!} - \dots $

For a discrete representation, values of $ u $ can be indexed at each discrete point using some index $ i $, so that $ u_i = u(x_0) $, $ u_{i+1} = u(x_0 + \Delta x) $, etc...

$ \left. \frac{\partial u}{\partial x} \right|_{i} = \frac{ u_{i+1} - u_{i} }{ \Delta x } - \left. \frac{\partial^2 u}{\partial x^2} \right|_0 \frac{\Delta x}{2!} - \dots $

Let's look at this last term...

Tells us two things:

1. For the difference approximation of the derivative to be accurate, we have to have $ \Delta x $ become really small.

2. We ALSO have to have $ \frac{\partial^2 u}{\partial x^2} $ be really small.

That means if we're trying to approximate a function with a large second derivative, we need an even smaller $ \Delta x $.

(Two images... one a nice 3rd or 5th order polynomial, the other the complex 100th order polynomial... "Just to make sure this concept makes sense.... which one has a larger second derivative? Which one requires a smaller $ \Delta x $?)

We didn't get an exact answer to the question "how big should $ \Delta x $ be, but we've got an idea now

What if the last term looked like this?

$ \left. \frac{\partial^2 u}{\partial x^2} \right|_0 \frac{(\Delta x)^2}{2} $

What does that tell us?

1. The value of $ \Delta x $ can be bigger to approximate this function

• What we're trying to do with it - i.e. discretize it (define it, explain it)

• Definition of limit

• Derivative is to continuous function what algebraic difference is to discrete function
  • i.e. imagine a limit for a discrete function...... it wouldn't turn into a derivative!
  • $ \lim_{\Delta x \rightarrow \delta} $ versus $ \lim_{\Delta x \rightarrow 0} $

• Taylor series
  • Expand a nearby point $ (x+\Delta x) $ about another point $ (x) $
  • Rearrange to put difference in terms of derivative
  • Truncation error, order of error