# Computational approaches to analyse numerical solutions

I apologise beforehand if the question isn't well defined or is too broad, I'm just having some difficulty finding the information I need. I also apologise if this should have been asked on stack overflow.

I have computed some numerical solutions using spectral methods to some different hydrodynamics problems that I'm interested in. The solutions are given as time series arrays. For example, at each time $t = 1, \dots, n,$ the solution for the velocity field $v$ is an $m \times m$ array. I would like to analyse my system to determine as much information about it as possible. For example, in my 2D vortex simulations, I would like to numerically determine the dominant behaviour of the system, or numerically find/predict any splitting or merging of vortices, or perhaps find relationships between the different variables of the system etc. The problem is, I don't actually know of many computational methods that I can use to determine these kinds of behaviour. Some things I do know we can use (and have implemented) are

• Autocorrelation functions (in both time and space).
• Singular value decomposition (to determine dominant behaviour).
• Analysing the spectra of the solution (radially averaged bins, FFT).
• Basic statistical analysis, analysing fluctuation data.
• Coupling tracer particles/dyes into the DE system to observe phenomena.
• Visualisation of the numerical solutions.

Obviously, comparing the numerical solution with the analytical solution is also a useful technique, though most of the time this isn't possible in hydrodynamics.

I was hoping that there were more numerical techniques available that I could learn and use to study my system. So I was wondering if someone could let me know of any papers or books that detail approaches to analysing numerical solutions, or could list down some methods that they know of. Any help would be greatly appreciated.

EDIT 02/12

I was given a review paper today by Jiang et al. which outlines nine numerical methods for detecing vortices, and also a paper by Jeong et al.. It is mostly these types of algorithms/computational methods that I am interested in using.

• I think that you want to analyze your results, but also verify them. You could look for the Method of Manufactured Solutions. Nov 30, 2017 at 15:43
• I would say that your steps seem interesting, but I would ad visualization of different variables as a main step, as well. Nov 30, 2017 at 15:47
• @nicoguaro Thanks, I'll look at the manufactured solutions stuff when I get a chance. I'm actually more interested in determining how, when and where coherent stuctures form in my solutions than in verification of the solution itself. If I have a time series with data points 'close enough' in time, I would have though (though I'm probably wrong) that hidden in the solution somewhere should be pattern that allows me to determine any underlying coherent structures, symmetries etc. that can be used to gain insight to the problem. Nov 30, 2017 at 16:11
• When analytical solutions are unavailable and the phenomenon of interest is too complicated to generate a suitable manufactured solution, it may be better to try to replicate published results similar to your case of interest.
– Paul
Nov 30, 2017 at 17:21
• It seems to me that you're interested in the physics of the result, not the properties of numerics or their convergence to (hopefully) the actual solution. As such, the post processing and analysis are dependent upon the problem itself, and what quantities are of interest. Perhaps some more detail on the specific problem is warranted here...
– user20857
Nov 30, 2017 at 20:50