Laplace's Demons

• GIF

  Animation of the discrete coarse plot with respect to increasing relaxation on the sin(xy) function.

• GIF

  Animation of the interpolation plot and respective error with respect to increasing relaxation on the sin(xy) function.

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  Discrete coarse plot of f(x,y) = sin(xy) with 0 relaxations.

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  Discrete coarse plot of f(x,y) = sin(xy) with 10 relaxations.

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  Discrete coarse plot of f(x,y) = sin(xy) with 100 relaxations.

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  Interpolation plot of f(x,y) = sin(xy) with 0 relaxations.

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  Interpolation plot of f(x,y) = sin(xy) with 10 relaxations.

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  Interpolation plot of f(x,y) = sin(xy) with 100 relaxations.

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  Discrete coarse plot of f(x,y) = x^2 with 1000 relaxations.

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  Interpolation plot of f(x,y) = x^2 with 1000 relaxations.

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  Attempted Algorithm for Polar Graph Solution Taking Advantage of Exact Value at Origin.

Inspiration for this project came from the one and only McGill Professor of Mathematics, Rustum Choksi. One group member is currently taking his course on PDEs and felt his passion when covering the Laplace equation. Harmonic functions encode symmetry like no other and the beauty of functions over discrete graphs whose values at any arbitrary node (not on the boundary) depends precisely on the value of every other node was irresistible.

What it does

Our program uses a relaxation method to numerically solve for the specific harmonic function over the unit disk in R2 with boundary value f(x,y) on S1. Here, f is an arbitrary continuous function that we can manipulate as an input. Additionally, our program determines the relative error at the origin because this is a value that can be solved for explicitly.

How we built it

We used a pascal algorithm with relaxation.

Challenges we ran into

We faced many challenges considering neither of us had ever done any numerical analysis. Additionally, one partner is still learning multivariable calculus.

Accomplishments that we're proud of

We are proud of all of it! It was quite rewarding.

What we learned

We learned pretty much everything we implemented on the spot.

What's next for Laplace's Demons

More hackathons!

Built With

• matlab
• python
• scipy