Publication Date


Document Type


Committee Members

Amir Farajian (Committee Member), James Menart (Advisor), Zifeng Yang (Committee Member)

Degree Name

Master of Science in Engineering (MSEgr)


PIC (Particle-in-cell) modeling is a computational technique which functions by advancing computer particles through a spatial grid consisting of cells, on which can be placed electric and magnetic fields. This method has proven useful for simulating a wide range of plasmas and excels at yielding accurate and detailed results such as particle number densities, particle energies, particle currents, and electric potentials. However, the detailed results of a PIC simulation come at a substantial cost of computational requirement and the algorithm can be susceptible to numerical instabilities. As processors become faster and contain more cores, the computational expense of PIC simulations is somewhat addressed, but this is not enough. Improvements must be made in the numerical algorithms as well. Unfortunately, a physical limit exists for how fast a silicon processor can operate, and increasing the number of processing cores increases the overhead of passing information between processors. Essentially, the solution for decreasing the computational time required by a PIC simulation is improving the solution algorithms and not through increasing the hardware capacity of the machine performing the simulation. In order to decrease the computational time and increase the stability of a PIC algorithm, it must be altered to circumvent the current limitations. The goal of the work presented in this thesis is twofold. The first objective is to develop a three-dimensional PIC simulation code that can be used to study different numerical algorithms. This computer code focuses on the solution of the equation of motion for charged particles moving in an electromagnetic field (Newton-Lorentz equation), the solution of the electric potentials caused by boundary conditions and charged particles (Poisson's Equation), and the coupling of these two equations. The numerical solution of these two equations, their coupling, which is the primary cause of instabilities, and the severe computational requirements for PIC codes make writing this code a difficult task. Solving the Newton-Lorentz equation for large numbers of charged particles and Poisson's equation is complex. This is the focus of this newly developed computer code. The second objective of the work presented in this thesis is to use the developed computer code to study two ideas for improving the numerical algorithm used in PIC codes. The two techniques investigated are: 1) implementing a fourth order electric field approximation in the equation of motion and 2) solving for the electric field, i.e. solving Poisson's equation, multiple times within a single time step. The first of these methods uses the electric fields of many cells that a charged particle may pass through in one time step. This is opposed to using only the cell of origin electric field for the particle's entire path during one time step. The idea here is to allow PIC codes to use larger time steps while remaining stable and avoiding numerical heating; thus reducing the overall computer time required. The second technique studied is utilizing multiple Poisson equation solves during a single time step. Typically, an explicit PIC model will solve the electric field only once during a time step; however, solving the field multiple times during the particle push allows particles to distribute themselves in a more electrically neutral manner within a single time step. The idea here is to allow larger time steps to be used without obtaining unrealistic electric potentials due to an artificial degree of charge separation. This eliminates instabilities and numerical heating. Explicit PIC codes have limits on how large the numerical time step can be before the electric potentials blow up. This work has shown that neither of these techniques, in their current state, are practical options to increase the time step of the PIC algorithm while ...

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Department or Program

Department of Mechanical and Materials Engineering

Year Degree Awarded