DOCC BLOGG

Heating a gas to a sufficiently high temperature result in electrons that leave the orbit of their atoms. The resulting material is called plasma (the fourth state of matter beside solids, liquids, and gases) which is simply a collection of a vast amount of positively and negatively charged particles (ions and electrons). It  is  easy  to  simulate  the  motion  of  such  a  charged  particle  by  using  Newton’s  law,  mx¨  =  F, where the force under consideration is the Lorentz force and arises when electric and magnetic effects act on charged particles. In a plasma, we have many particles (usually on the order of 1019), and therefore it is impossible, even with the largest supercomputers, to simulate them all. Nevertheless, it is not interesting at all to know the trajectory of all the single particles. It is more important to understand the behavior of averaged quantities as density or energy, on which many particles have an impact. Therefore, simulations that use much fewer particles than encountered in reality already give a qualitatively good description of the evolution of plasma. However, this task is still challenging since the moving charged particles generate a self-consistent electric field that impacts the force acting on them. Simulations conducted by such particle-based methods are called Lagrangian methods.

A different strategy to simulate the evolution of plasma is to consider its density directly. To this end, a density function is introduced that tells us how many particles exist at a given time at a specific position in space having a certain velocity, f (t, x, v). By doing numeric simulations, we have to fix locations in space on which we want to know and simulate this density. Therefore a regular fixed grid  is often chosen (see the left picture in Figure 1). The evolution of f can then be simulated by solving a kinetic equation, which can be considered as a generalization of Newton’s law mentioned above, where the electric field depends on the density. Such grid-based schemes are called Eulerian methods. Figure 1 illustrates the different grid strategies.

Both of the methods have strengths and drawbacks. First, since computers are doing the work for us, let us look into the memory requirements for the different approaches. For particle-based methods, the required amount of memory is determined by the number of particles used. Thus the memory requirement scales linearly with the number of particles used. Grid-based methods instead discretize the entire region under consideration. Since the density depends on space and velocity, 6d problems appear, which implies that the amount of required memory scales with a power of six. Using only 200 grid points in each dimension implies that more than 500 terabytes of memory are necessary to store the density function. The massive memory usage is the biggest and most important drawback of Eulerian methods compared to Lagrangian methods. Second, more used particles in the simulation give better results. The same is true for grid-based methods, which means that the more discretization points used, the better the density is approximated. However, the introduced error of particle-based schemes reduces much more slowly compared to the error of Eulerian methods. Moreover, regions with few particles suffer from numerical noise and inaccuracies. Third, explicit Eulerian methods suffer from step size restrictions due to numerical instabilities. Hence, in the simulation, just small time steps can be taken such that the simulation will not crash. Lagrangian methods do not suffer from such restrictions and can take much larger timesteps. Consequently, the overall simulation time of Lagrangian schemes can be significantly lower, which is beneficial.