Can someone explain how this solves a differential equation, and why the differential equation is on a grid?
JuneBot
@cube Suppose I want to calculate the electric fields generated by some extended object that doesn't have any nice symmetry properties. Then I have to solve Maxwell's equations numerically, by pretending space is actually a finite lattice. So the equations are on the grid because the grid is a model of space. This particular method is called relaxation, and you can find the formal structure here. Under the hood, relaxation is really just another approximation method that relies on the fact that the derivative of a function at a point is a good approximation for that function close to that point.
This is kind of a small example though, and you could run a pretty accurate relaxation for a fairly complex scenario right on your laptop, without needing any parallelism at all. There are currently active areas of research in physics that discretize spacetime on a lattice and perform more abstract calculations using Monte Carlo methods. A quick search will give you dozens of articles, and some of these people are even concerned with parallel hardware. This is a much more relevant example for our class, and you can see in section 2.3 that they chose a message passing model! I believe this is the standard choice for highly parallel physics these days. If you want to know more, there's actually a group at CMU working on precisely this type of computation.
Can someone explain how this solves a differential equation, and why the differential equation is on a grid?
@cube Suppose I want to calculate the electric fields generated by some extended object that doesn't have any nice symmetry properties. Then I have to solve Maxwell's equations numerically, by pretending space is actually a finite lattice. So the equations are on the grid because the grid is a model of space. This particular method is called relaxation, and you can find the formal structure here. Under the hood, relaxation is really just another approximation method that relies on the fact that the derivative of a function at a point is a good approximation for that function close to that point.
This is kind of a small example though, and you could run a pretty accurate relaxation for a fairly complex scenario right on your laptop, without needing any parallelism at all. There are currently active areas of research in physics that discretize spacetime on a lattice and perform more abstract calculations using Monte Carlo methods. A quick search will give you dozens of articles, and some of these people are even concerned with parallel hardware. This is a much more relevant example for our class, and you can see in section 2.3 that they chose a message passing model! I believe this is the standard choice for highly parallel physics these days. If you want to know more, there's actually a group at CMU working on precisely this type of computation.