This is Nathan. Thanks for the question regarding the computational methods used in plasma physics/tokamak turbulence simulation. Let me try to answer your question. So my work actually focuses on turbulent transport model validation (the comparison of experimental measurement with computational models) in the core of tokamak plasmas. Fundamentally the difficultly with plasma simulation comes down to the fact that we have a large number of particles (~1x10^20 m^-3) which due to the long range nature of the electromagnetic force, affect the movement of every other particle in the plasma, at every point in time and operate on a wide range of time and spatial scales. Now there are some simplifications to that statement but I wont go into those here. In theory, the Newton-Maxwell set of equations, that is the combination of Newton's laws with Maxwell's equations can give you a complete description of all phenomenmom in the plasma. However, given the large number of particles and the long range nature of the forces involved, this problem is computationally intractable. The first step of simplification is taking a statistical approach to describing the particles in the plasma, that is, describing each population with a distribution function and tracking the evolution of the distribution function. This set of equations is known as the Maxwell-Boltzmann set of equations. However, this is a 6-D set of equations and we require some more simplifcation. To do this we basically elminate the motion of very fast time scales from this set of equations and say we are look at spatial scales which are relevant for microturbulence which drives the transport of heat, particles, etc. in the plasma. The resulting equation is called the gyrokinetic equation and it is thought to contain sufficient physics to simulation the microturbulence which is present in pretty much all plasmas. This is the cutting edge model for turbulence simulation in tokamaks. In these simulations you can see the development of turbulent eddies and structures which result in the transport of heat and particles out of the system on a time scale much shorter than that from purely collisions.
Now, as you correctly noted, measurements in high temperature plasmas are indeed quite difficult to make and it is only in the last 10 years that both the models and the measurements have become accurate enough to make any meaningful comparision between the two. In other words, we are just now to the point where we can validate these models through comparison with experiment. To make things more difficult, the temperature profiles in plasmas are often observed to be "stiff". This means that the normalized gradients of these quantities are not easily changed. This arises from the fact that the plasma microturbulence is driven by free energy in the temperature and density gradients. As a result, if the gradient increases, the level of turbulence in the plasma does as well and this results in larger eddies and therefore larger loss of particles and energy from the system. This loss of energy tends to flatten the profiles and reduce the gradient. Therefore there is a critical value at which the turbulence is found to "turn on" and there is a profile shape which the plasma tends to relax to. This makes simulation difficult because it increases the sensitivity of the predicited heat/particle flux to the background plasma gradients. Since the gradients are derivatives of quantities which we measure, they are much more difficult to determine with great accuracy. My own work involves assessing the experimental error in many of the quantities that are driving plasma turbulence. With a proper assessment of the uncertainity in the measurement, I am able to deduce the sensitivity of the model's predictions to the uncertainity in the measurement and make a more meaningful statement of the model's ability to simulate experiment.
I am sorry if that was a bit too rambling, but hopefully I at least partially answered your question. If you are interested in learning more about simulation of plasma turbulence and the gyrokinetic model, I would refer you to a review paper by X. Garbet which was published in the Journal Nuclear Fusion, 50, (2010). If you have access to this journal through your institution, you can find it here: doi:10.1088/0029-5515/50/4/043002