euleretal (Euler et. al.)

Overview

This is an interactive visualization tool for exploring the qualities of some discrete methods for solving second order differential equations. Practically, this is about calculating the trajectory of rigid bodies (masses) accelerated in different kinds of force fields. (But let's forget about masses and forces and simply talk about acceleration fields.)

The idea is to compare the precisions of different algorithms of, for example the Euler and Runge-Kutta family, and their suitability for different force fields.

This is work in progress, and in a very early stage (pre-alpha). I am still experimenting and refactoring the code structure. There is not much you can do, yet, but it is already possible to compare the precision of a few discrete integration methods. You can test the application online.

Disclaimer

I am not a mathematician or expert in this area. (If I was, I wouldn't have had the need to write this tool in the first place.) It is likely that I use non-standard or wrong names for some of the methods, and I might implement them in the wrong way. If you have suggestions, do not hesitate to create an Issue, Discussion, or send me an email.

Why?

I wanted to write a 3D-Simulation of particles being attracted or repelled by other particles or objects in a simple world. I quickly found that my particles diverted from the paths I would have expected them to take. This became obvious when two masses came too close to each other, or when I created a “wall” which was supposed to repell any particles with a force which should asymptotically approach infinity (I can hear you loughing!).

So I read about different methods like Euler or Runge-Kutta (todo: citation needed), but it was hard for me to understand all the details or follow the theoretical arguments. I wanted to develop an intuitive idea about why some Runge-Kutta methods make sense, and I found myself drawing lines on paper. Alas, these lines did not help me because I was not able to draw them with sufficient precision in order to understand whether my ideas would work out.

…and then there was the moment where I realized that the Euler method was plainly wrong when it comes to second order integration:

v' = v + a(s) dt
s' = s + v' dt

Inserting the first formula into the second, we get:

s' = s + v dt + a(s) dt²

But I knew from high school physics that the following is correct when a is constant:

s' = s + v dt + ½ a dt²

This means that the Euler method is not even correct for the trivial case of constant acceleration — WTF! The literature I found so far, often made a statement similar to this one on Wikipedia: “Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations.” Well, while this may be true in theory, I now know that I can loose big in practice if I apply first order methods to my second order problem.

Finally, I came up with some formulas on my own. I am 100% sure that others have used them before, but I do not have the mathematical background to name them. This is where my frustration turned into curiosity and fun. I wanted to see the formulas in action, so I started this little project.

License

Licensed under either of

Apache License, Version 2.0 (LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)
MIT license (LICENSE-MIT or http://opensource.org/licenses/MIT)

at your option.

Contribution