So I’ve moved on to curved lines now, in case you’re wondering. (See previous posts.) The last several days (the research parts) have been taken up with more computations. A lot of the time has been spent calibrating the programs, and trying to assess and understand and characterize the inevitable errors that show up, by running the programs and checking the resulting plots of data points against expectations shaped by hand calculations. Calculating on the train to and from work, I’ve filled several pages of my small notebook with computations, alongside sketches of some of my surroundings as usual (people mostly). As a result (fingers crossed) I think I’ve now understood all the key aspects of the results I’ve been getting, and have good numerical control of things. To get such control, I’ve had to push the error tolerance and the size of the grid of points I’m computing on to regimes where I’m back again to waiting for the better part of an hour for each data point. (One sets up the problem on the computer by making continuous variables, such as space and time, into discrete ones, forming a grid. The problem is then to use various algorithms to solve the equations by computing the values of the variables of interest (numbers) at each grid point, to within certain error tolerances. The finer the grid, and the smaller the error tolerance, the closer it is to the real system of interest.) One version of the program has parameters set so that a point takes several hours. So it will be a while before I get all my data. [Update 9th May: I’d actually not fixed one source of error when I wrote that yesterday, but the good news is that now I have found and fixed that (using an idea I had while on my way to the dentist) so all is well.]
This leaves me time to think about what sorts of things I can deduce by hand. While one cannot solve the equations fully due to their (aforementioned) severe non-linearity, there are often regimes of the problem where one can simplify, and then using analytic (i.e. non-numerical) means solve the system. This often allows for useful checks on the larger computation in some circumstances.
I’m chatting about all this since I sometimes get the sense that people are not so familiar with some of the ins and outs of research in my area, and I’m sometimes asked how I use computers in my work. So there you have it. A little bit.