Computational Methods Junior Honours : Checkpoint 2

Assessment: MarkingCP2.htm

Starting point : Complex class

• Download the java file Complex.java

• Read it

Complex.java defines a public class for complex variables.

• Download SlowFourierTransform.java

• Compile them

javac Complex.java SlowFourierTransform.java

Your task: Discrete Fourier Series & linear PDE’s

Make yourself familiar with the discrete Fourier series described in the background material and the lecture

This is normally, in computing, called the discrete Fourier transform, but it is actually a special case of the Fourier series you are covering in Physical Mathematics.

This is NOT the true Fourier transform used in Mathematics and Physics (which is a bit more complicated).

I will at times follow the “computing” convention and abuse “transform” because the “FFT” is the world famous “Fast Fourier Transform”, even if it should be “FFS”.

The discrete Fourier series in one dimension of F_x with N points x=0… N-1 is defined as:

f_k = 1/sqrt(N) Σ_x=0..N-1 e ^{i 2 π k x / N} F_x

The discrete Fourier series in two dimensions of F_x,y with N² points x=0… N-1 ; y=0…N-1 is defined as:

f_k,j = 1/N Σ_x=0..N-1 Σ_y=0..N-1 e ^{i (2 π k x / N)} e ^{i (2 π j y / N)} F_x,y

The inverse is

F_x,y = 1/N Σ_{k=0..N-1 Σj=0..N-1} e ^{-i (2 π k x / N)} e^{-i (2 π j y / N)} f_k,j

Or

F_x,y = 1/N Σ_k=0..N-1 Σ_j=0..N-1 e ^{-i ω.x f}_k,j

Where

ω = (ω_k, ω_j)

ω_k = 2 π k / N

and

x= (x,y)

Observe

▽² F_x,y = (d²/dx² + d²/dy²) F_x,y

= - 1/N Σ_k=0..N-1 Σ_j=0..N-1 (ω_k²+ ω_j² ) e ^{-i ω.x} f_k,j

= - 1/N Σ_k=0..N-1 Σ_j=0..N-1 |ω|² e ^{-i ω.x} f_k,j

Now, Poisson’s equation relates the electrostatic potential to the electrostatic charge density

E = - ▽ V

▽. E = ρ / ε₀

▽²V = - ρ / ε ₀

Supposing we represent V_x,y and ρ _x,y via their Fourier series. This differential equation becomes

true for each Fourier component.

- |ω|²V_k,j = - ρ_k,j / ε ₀

Thus,

V_k,j = ρ_k,j / (|ω|² ε ₀)

Taking ε₀=1, if we know ρ(x,y) we can compute V(x,y) by applying the derivative to the Fourier series, and then inverting this:

V_x,y = 1/N Σ_k=0..N-1 Σ_j=0..N-1 e ^{-i ω.x} ρ_k,j /|ω|²

• Complete the SFT method in SlowFourierTransform.java

• Note you can compute e^iθ as

• Complex phase = new Complex(Math.cos(theta),Math.sin(theta));

• Comment SlowFourierTransform.java in your own words and be prepared to explain it

· Note, the charge density is set up in method coulomb() to be a single point charge

• How is the divide by zero avoided & what does this mean in terms of physics
• Be particularly prepared to explain the method coulomb to your demonstrator

• Strictly speaking, this isn’t really a coulomb potential – as this is a point charge in a 2d equation

it is equivalent to a infinite line of charge in 3d. Couldn’t think of a better name though…

We could easily do 3d but it is harder to visualize.

As we’re in 2d the field spreads out as the circumference of a circle – 1/r (not the area of a sphere 1/r²)

This means the potential looks like log r.

• Compile and run SlowFourierTransform

• javac SlowFourierTransform.java
• java SlowFourierTransform > coulomb.dat

• Use gnuplot to view the output

· gnuplot

• At the prompt type

• splot ‘coulomb.dat’ using 1:2:3 with lines

• This told gnuplot to plot x,y,F(x,y) data using columns 1,2,3 of the file

• Type

· set contour

• replot

· Harder: Add a new method void capacitor()

• This should be just like coulomb() but use a different charge density.
• Set the charge to be
• +1 on the line from (28,24) to (28,40)
• -1 on the line from (36,24) to (36,40)

• Why does this look like a capacitor?
• Change main to call your new method
• Plot the potential and be prepared to explain it to your demonstrator
• What does the electric field look like

END

· Congratulations – you can now solve any electrostatics problem using Fourier series!

· This includes the edge effects around capacitors, and no Gaussian pill boxes in sight!