Article: Computational Plasma Physics
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First, this review article gives an insight into the history of the development of plasma physics and what subfields arose during the development of the field, because it is of the most importance to understand what problems can be solved using such complicated object as plasma. Next, four main mathematical models of plasma are introduced in the article. They play the key role in the plasma physics, because they set differential equations, which specify scales on which model is correct and the mathematical complexity of the object. Computational methods are discussed later on in the article in order to give the reader all necessary information about algorithms which can be used to simulate a chosen mathematical model. At the end, one of the most commonly used computational methods, the Particle-in-Cell method, is described a little more in detail. Not only does this method plays an important role in today’s explorations, it also has implications for future prospects due to improvements in computing power and architecture.
History of plasma physics
Originally, the term plasma meant “cleared blood” (oxforddictionaries.com, 2017). Irving Langmuir, the Nobel prize-winning chemist and physicist, was the first person who used this term to describe an ionized gas. He provided an analogy between physical and biological plasmas, saying that “the way blood plasma carries red and white corpuscles by the way an electrified fluid carries electrons and ions” (Fitzpatrick, 2014). The theory of plasma was developed when Langmuir tried to extend the lifetime of the filament of a tungsten-filament light-bulb. After Langmuir’s experiments, the theory of plasma spread in five other directions: electromagnetic wave propagation through nonuniform magnetized plasmas, magnetohydrodynamics, thermonuclear fusion, space plasma physics and laser plasma physics (Fitzpatrick, 2014).
Mathematical models of plasma
Kinetic models are the most fundamental way to describe plasmas. It uses Vlasov equation, or more precisely Vlasov-Maxwell system of equations:
There are a lot of different fluid models of plasma, but one particular set of models are of the utmost interest to researchers, because they are “simple” relative to other models: they consider only one fluid, and at the same time they contain a lot of important macroscopic properties of a plasma: the plasma velocity field, current density, mass density, and the plasma pressure (Callen, 2003). Magnetohydrodynamics (MHD) is the simplest model of such kind, which contains equations of one-fluid model dynamics coupled with the Maxwell equations for the electromagnetic fields (Callen, 2003).
Computational methods of plasma physics
The first and the easiest to understand approach to simulate plasma on a computer is to use N-body model. This technique allows computing full trajectories of each plasma particle -- “the first-principle dynamics” (Miloch, 2014). Such a detailed description of plasma particles movement affects exterior electric and magnetic fields and, as a consequence, these fields must be updated on every step of a simulation. Because each particle in the system interacts with every other particle in the system, the computational complexity of solving the N-body model is, which is practically unsolvable if a number of particles are too big (Miloch, 2014).
Another type of computational methods is based on solving the Vlasov-Maxwell equation, which is a nonlinear hyperbolic partial differential equation. The solution of this equation is a “Hamiltonian flow in a phase-space” (Miloch, 2014). There are different algorithms which allow solving numerically the kinetic plasma model: “e.g., semi-Lagrangian method, finite volume methods” (Miloch, 2014), though all of them are very computationally complex.
Fluid models approximate plasma dynamics on large scales and describe the evolution of macroscopic properties: density and pressure. Unlike the difficult Vlasov-Maxwell equation which can be solved only using complicated numerical algorithms and linearisation, MHD equations, which can be rewritten in the form of a first order hyperbolic partial differential equation, can be integrated using well-known Finite Difference methods like “Lax, Lax-Wendroff, leap-frog” (Miloch, 2014) as well as different Finite Volume algorithms. Finite Difference methods are divided into two categories: explicit and implicit algorithms. Explicit schemes are less computationally complicated, but they require a very strict Courant condition for a stable solution. On the other hand, implicit schemes are computationally stable, but they are more difficult in terms of programming and require more computational power. Therefore, researchers usually use hybrid explicit-implicit schemes as a “compromise between accuracy and numerical efficiency” (Miloch, 2014).
The systems that are of the interest to researchers are mostly very large in terms of the number of electrons and ions. To be able to simulate the dynamics of such systems, “super-particles” or “computer particles” are used instead of real particles. A super-particle is a computational particle which contains many real particles; it may be millions of electrons or ions in the case of a plasma simulation, therefore unlike real particles which are represented as mathematical points, super-particles should have a physical radius, because they contain a lot of real particles. Because the Lorentz force depends only on the charge to mass ratio, it is allowed to replace a number of particles with one super-particle with the same charge to mass ratio, and it will follow the same trajectory as the system of real particles (Filipič, 2008).
The Particle-In-Cell method
It’s technically impossible to solve the equations of motion and the system of Maxwell equations in the continuum space. In order to solve this problem the physical volume is divided into cells, creating a mesh, and the electromagnetic fields are getting computed only in the intersection points of this mesh (Filipič, 2008). Each cell has a local charge density which is the “number of particles in the cell divided by the volume of the same cell” (Filipič, 2008). Close interactions between particles have a low importance for a macroscopic problem, so electromagnetic fields are computed using local charge densities to compute potential and then the fields itself rather than “summing contribution of each particle” (Filipič, 2008). It’s also important to keep cell’s length in a very concrete range: it should be small enough, so the charge density separation effect can be noticed in computed fields, and at the same time large enough, so spatial cells will be bigger than the size of super-particles (Filipič, 2008).
The actual algorithm for performing a simulation consists of several steps. After reading the initial conditions, charge densities should be obtained in order to proceed with the integration of field equations. Next, the interpolation of electromagnetic fields from the mesh to the particles’ locations is performed in order to calculate to what extent “each individual grid point affects each particle” (Filipič, 2008). The last step is the integration of the equations of motion and changing the positions and velocities of each particle. After that the procedure repeats (Filipič, 2008).
Conclusion
Bibliography
Escande, D., Elskens, Y., Doveil, F. (2012) Basic microscopic plasma physics unified and simplified by N-body classical mechanics, eprint arXiv:1210.1546.
Filipič, G. (2008) "Computer simulations" in Principles of ”Particle in cell” simulations, University of Ljubljana, Faculty of mathematics and physics, p. 4-5.
Fitzpatrick, R. (2014) “Introduction” in Plasma Physics, CRC Press, Boca Raton, FL, p. 5-8.
Kalitkin, N., Kostomarov, D. (2006) Mathematical models of plasma physics (review), Matem. Mod. (vol. 18) Number 11, p. 67–94
Miloch, W. (2014) “Numerical Methods for Plasma” in Plasma Physics and Numerical Simulations, University of Oslo, p. 7-9.
Plasma, https://en.oxforddictionaries.com/definition/plasma, 2017.
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