Vlasov equation can be treated numerically with the test particle method.
From Vlasov one can derive a set of hydrodynamic equations which again can be tackled with the so called Smoothed Particle Hydrodynamics in which fictive particles are used to reconstruct the density.
Why the equations of motion differ between this two methods?
The density is constructed in the same way.
Yes! It is used a spectral decomposition of distribution function in terms of gaussian function centred on some time variable positions
I am assuming that by "test particle method" you mean something akin to the Particle In Cell (PIC) method. This method samples the complete phase space with test particles. It is a kinetic approximation. Each particle is characterised by mass, charge, position, velocity (or momentum) and a particle weight. The equations of motion of the test particles are simply Newton's equation of motion. If you want to know what the density, average velocity, temperature, etc is at a given location you need many particles at a given location (inside a cell). For example the temperature is calculated from the spread of velocities of many particles within a cell.
Smoothed Particle Hydrodynamics, on the other hand, is based on the fluid equations and does not capture any effects due to non-Maxwellian distribution functions. Each particle in this approach also carries information about the internal energy which allows one to calculate the temperature and pressure at the location of the particle. You don't need many particles at any given location to calculate the macroscopic quantities. In this way a particle represents a fluid element.
Ok, H, Schmitz, thank you for your answer. I am aware about all these things. I guess, now the question is if you think that from numerical point of view the only difference is due to the fact that you need less particles in SPH since the momentum space is not sampled?
Also, should I expect that in the N goes to infinity (N is thr number of pseudoparticles) limit, the contributions from internal energy (pressure) cancel each other in the force calculations?
Hi Dragos Palade,
I would understand that SPH and PIC are two completely different numerical approaches. The differences between them can't simply be reduced to the fact that you need less particles for SPH. SPH models the hydrodynamic equations, PIC models the kinetic equations. PIC evaluates fields on a numerical grid, SPH is a grid-less method where forces are calculated from neighbouring particles only. In SPH interactions are limited to short ranges determined by the smoothing kernel. In PIC you can have long range interactions, e.g. in the electrostatic limit you solve Poisson's equation from the charge density on the grid. These are just a few of the differences between the two methods.
The Vlasov-Poisson system (for example) can be reduced to a hydrodynamic system of equations, from which, if you have the "correct pressure tensor", you should be able to reproduce the same results regarding the density and the current in the system.
Such an example in which hydrodynamic equations are solved with SPH can be found in [http://info.ifpan.edu.pl/ON-2/on26/pubs/PhysRevA_71_043202.pdf].
Further, from the position of pseudo-particles from SPH one can construct the density on a grid, solve Poisson equation, and from the interpolated potential to compute the forces on each particle (sometimes, this approach can be faster than tree-based methods).
Therefore, I don't see such dramatic differences between the kinetic approach and its corresponding hydrodynamic approach. As I said, is my understanding that a correct knowledge about the dependence between the pressure term and density would pave the way towards equivalent results between Vlasov and Hydrodynamcs.
Yes, you can integrate the moments of the Vlasov-Poisson system to obtain the hydrodynamic system of equations. In many cases, however, it is not possible to find an good closure of the system. In these cases the hydrodynamic equations are inadequate to describe your plasma and the kinetic equations must be used. Thus, the hydrodynamic equations and the kinetic equations represent two different models of a plasma.
Naturally this means that a code simulating the hydrodynamic equations can't be directly compared to a code simulating the kinetic equations. That said, here is a last example of how the two codes fundamentally differ. Imagine you have many particles close together and these particles can be split into two groups with very different velocities. In SPH this would indicate a sharp transition of the velocity such as in a shock. In PIC, on the other hand, this just means that your distribution function has two populations such as you would find in simulation of the two stream instability.
Nice example!
Still, you cannot deny that a principial knowledge of the pressure should give us the same results. Therefore, a good numerical representation should reflect in results on density and current.
Don't you agree?
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