Taylor Expansions in PDEs

Ever wondered what the uses for taylor expansions are in the field of differential equations? no? well you should, its rather facinating…

First, what is a taylor expansion? well, basically, it says if youre trying to evaluate a function at a point that’s “close enough” to a point you already know you’ll be able to represent this slight difference as an infinate series:

Or if youre more comfortable with summation notation:

Here \(O(h^3)\) represents an arbitary function of order \(h^3\) (basically just a function in \(h\) with the smallest \(h\) term being \(h^3\)), this is important and we will use this in a bit…

So how does this relate to a PDE? Well, we can estemate derivatives of a functon using this method, right? (as long as everything is nicly behaved) take eqn (1) above, and rearange it,get fid of a few terms & this will give you:

or setting \(F’(x)\approx f’(x)\)

From here we can see that we can somewhat accuratly calculate a derivative of a function using the evaluation at 2 points that are “close enough” together (this can be done in both directions by using \(-h\) insted). That’s cool, but now what?

mesh grid

See this grid? This is a plane where a PDE lives. We can solve the PDE at a point on this plane if we use the above fomulas in a clever way. Imagine were ust working on the line \(y=0\) and we want to work out \(f’(0.3)\) where \(f(x)=x^2\), how do we do this?

  1. Remember that we can aprox \(f’(x)\) using (4).
  2. Set \(h\) to be 0.1, or the spacing between the grid points.
  3. Then we get
  4. Which we get \(f’(0.3)\approx F’(0.3)=\frac{(0.4)^2-(0.3)^2}{0.1}= 0.7 \approx 2(0.3)\)
  5. And we have our approxamate answer at \(x=0.3\)

We can even make this more accurate by taking our expansion in \(+h\) and finding the difference from our expansion in \(-h\). essentially taking the difference between the 2 points either side of the position, \(x\), we’re looking at. we get:

Forward: \(f(x+h)=f(x)+hf’(x)+\frac{h^2}{2!}f’‘(x)+O(h^3)\)
Backward: \(f(x-h)=f(x)-hf’(x)+\frac{(-h)^2}{2!}f’‘(x)+O(h^3)\)

so we can take the difference: or setting \(F’(x)\approx f’(x)\)

Its a much nicer result beacuse our \(O(h^2)\) term well decay faster if we make the difference, \(h \rightarrow 0\). We can show that the error we get is much smaller if we try again, so lets do the exaple of \(f’(0.3)\) where \(f(x)=x^2\) again:

  1. With \(F’(x)\approx f’(x)\) we can use eqn (5).
  2. Subbing everything in we get \(F’(0.3)=\frac{f(0.3+0.1)-f(0.3-0.1)}{2(0.1)}=\frac{(0.4)^2-(0.2)^2}{0.2}=0.6\)
  3. TADAAA

Using this 2 sided approach is called the centered difference approximation and is much more accurate, it should be used if possible. Hint: if youre trying to calculatean approx for \(f’(x)\) and \(f(x\pm h)\) dosent exist, then a problem has been encoutered and must be solved. There are ways of solving these we will come on to.

But first I hear you ask “Toby, is there a way of doing this for approxamating a second derivative?” to which I reply: “why yes, its easy enough, just rearange the taylor expansion for the particular derivative you need!”. I have lied to you though, it is not “easy enough”; with a simple push however it can be.So what are we really looking for? We want to find this approximation to the second derivative, or \(F’‘(x)\approx f’‘(x)\). And we want this \(F(x)\) as a linear combination of the points \(f(x-h),\ f(x),\ f(x+h)\) or, in a more verbose defintion:

which then gives, after expainding these points out, or using both the farward and backwards expansions given above: \(F’‘(x)=A\big{[}f(x)+hf’(x)+\frac{h^2}{2!}f’‘(x)+O(h^3)\big{]}\)
\(\qquad \quad +B\,f(x)\)
\(\qquad \quad +C\big{[}f(x)-hf’(x)+\frac{(-h)^2}{2!}f’‘(x)+O(h^3)\big{]}\)

Where collecting terms in this expansion provides a nice set of coefficents to solve for:

We want to set \((A+B+C)=0\) , \((A-C)=0\) and \(\frac{h^2}{2!}[A+C]=1\) then the rest of the combos make up the error, so they can be ignored. Solving all of these we can just set:

  • \(A = C = \frac{1}{h^2}\) by the second two eqns.
  • \(B = -(A+B) = -\frac{2}{h^2}\) by the first.

These coeficients allow the approximation for \(f’‘(x) \approx F’‘(x)\) to be:

The motive for taking centered approximations can be seen below. Its obvious the red line has a gradient closer to \(f(4)\) than the other two lines.

aprox-graph

From this point on we can increase the dimension of \(f(x)\) to be a function of 2 variables: \(f(x,y)\), and we can expand \(f\) in both \(x\) (as we did before), and \(y\) (by using the spacing of \(\pm k\) insted of \(\pm h\)). Then we can calculate partial differentials in the same manner; expand the taylor series and rearange for an approximation. Only this time insted of arbitary spacing we use the concept of creating a mesh with points spaced out over our domain this is known asdiscritizing the domain(the grid above is a prime example). This then gives:

and

Note that the partial differentials have are actually evluated at a given \(x\) and \(y\) so it may be more obvious whats going on using: \(\frac{\partial f(x,y)}{\partial x} = \frac{\partial f}{\partial y}\mid_{x,y}\)