Posts about simulation
Better Numba calculation of inter-particle distance matrices
Recently, I've been looking for efficient ways to compute a distance matrix in Python. I'm deliberately trying to implement a naive n-body simulation so as to find optimized ways of calculating those, as practice. Let's do that using Numba.
Post mortem for my engineering thesis code, PythonPIC
I'm giving a presentation on this less-than-glorious subject on Friday, so I figured, hey, it might be a nice time to write a summary of what that old repository on my GitHub page is. In a single video:
Admittedly, this post is going to be rather personal - this messy little code was basically my life for a few hundreds of hours.
On the recent "On the Boris solver in Particle-in-cell simulations" paper
I recently came across a pretty cool paper by Zenitani and Umeda named "On the Boris solver in particle-in-cell simulation". There are many splendid descriptions of the Boris solver on the Internet, so while I would rather not duplicate them, here's a brief overview. In PIC simulations, the Boris solver (or pusher) is the usual algorithm of choice for moving and accelerating particles in given electric and magnetic fields.
You may wonder, since the equations of motion are ordinary differential equations, what's wrong with using the usual Runge-Kutta 4 solver? As it turns out, that one has a pretty major flaw. It has great accuracy for short term calculations, but over time your particle's motion will lose energy. This is a deal breaker for periodic motion, and simulations of, for example, plasma waves need to conserve that energy to provide accurate results.
Boris came up with his solver in the 1950's, and in a single sentence: the algorithm splits the acceleration via electric field into two parts and sticks a rotation about the magnetic field between them. This turns out to conserve energy and will probably come up again on this blog as I read more about symplexicity.
scipy.integrate.solve_ivp and makeshift Poincaré sections of the Rossler attractor
I've just stumbled upon a relatively recent addition to scipy: integrate.solve_ivp, which looks amazing for the simulation of dynamical systems and solving equations where you'd like to detect discrete events occuring (say, collisions). Let's a look at what it's all about, and then use it to simulate the Rossler attractor!
rossler_attractor
Quantitative data analysis of the 2D Ising model
Last time, we made a neat little implementation of a 2D Ising model simulation, and we showed that it looks reasonable. Well, that it might, but we can't be certain of that! I know I said that next time we would, er, %timeit, put it on the GPU and make it GO FAST, but perhaps it's a better idea to start with some data analysis first, making sure the result we're getting are quantitatively what we would like them to be - physical.
Parallelizable Numpy implementation of 2D Ising model
In the next few posts I'd like to discuss a fun project I've been procrastinating things with recently - a parallelizable (up to the GPU level) Python Numpy implementation of the 2D Ising model, which I won't introduce at length here because it's been covered really well in multiple places out there and I'd rather not repeat them.
The gist of it is that we have a grid of discrete spins represented by integers, +1 for spin up and -1 for spin down. Each spin interacts with its nearest neighbors via a sum of products of that spin's value and the neighbor's value, times minus a positive constant J. The minus is there so that spins pointing the same way decrease the total energy and spins pointing in the opposite direction increase it.