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Lorentz boosted frame simulation technique in Particle-in-cell methods

  • Author(s): Yu, Peicheng
  • Advisor(s): Mori, Warren B
  • et al.

Accelerators at the energy frontier have been the tool of choice for nearly a century for unraveling the structure of matter, space, and time. Today's accelerators are the most complex and expensive tools for scientific discovery built by humans. The capability of these accelerators has increased at an exponential rate due to the development of new accelerator concepts and technology. The capability of existing accelerator technology has plateaued, so that a future accelerator at the energy frontier will be so large and expensive that it is not clear it will be built. On the other hand, plasma based acceleration has emerged as a possible alternative technology with much recent progress in theory, simulation, and experiment. In plasma based acceleration intense short-pulse laser, or particle beam excites a plasma wave wakefield as it propagates through long regions of plasma. When a laser is used it is called laser wakefield acceleration (LWFA), and when a particle beam is used it is called plasma wakefield acceleration (PWFA). Simulations have contribute greatly to the recent progress by providing guidance and insight for existing experiments, and for permitting the study of parameters beyond the current reach of experiments. However, these simulations require much computing resources. Therefore, alternative numerical techniques are desired, and in some cases are needed.

In this dissertation, we systematically explore the use of a simulation method for modeling LWFA using the particle-in-cell (PIC) method, called the Lorentz boosted frame technique. In the lab frame the plasma length is typically four orders of magnitude larger than the laser pulse length. Using this technique, simulations are performed in a Lorentz boosted frame in which the plasma length, which is Lorentz contracted, and the laser length, which is Lorentz expanded, are now comparable. This technique has the potential to reduce the computational needs of a LWFA simulation by more than four orders of magnitude, and is useful if there is no or negligible reflection of the laser in the lab frame.

To realize the potential of Lorentz boosted frame simulations for LWFA, the first obstacle to overcome is a robust and violent numerical instability, called the Numerical Cerenkov Instability (NCI), that leads to unphysical energy exchange between relativistically drifting particles and their radiation. This leads to unphysical noise that dwarfs the real physical processes. In this dissertation, we first present a theoretical analysis of this instability, and show that the NCI comes from the unphysical coupling of the electromagnetic (EM) modes and Langmuir modes (both main and aliasing) of the relativistically drifting plasma. We then discuss the methods to eliminate them. In EM-PIC simulations of plasmas, Maxwell's equations are solved using a finite difference form for the derivatives in real space or using FFT's and solving the fields in wave number space. We show that the use of an FFT based solver has useful properties on the location and growth rate of the unstable NCI modes. We first show that the use of an FFT based solver permits the effective elimination of the NCI by both using a low pass filter in wave number space and by reducing the time step. We also show that because some NCI modes are very localized in wave number space, a modification of the numerical dispersion near these unstable modes can eliminate them. We next show that these strategies work just as well if the FFT is only used in the plasma drifting direction and propose a hybrid FFT/Finite Difference solver. This algorithm also includes a correction to the current from the standard charge conserving current deposit that ensures that Gauss's Law is satisfied for the FFT/Finite Difference divergence operator.

However, the use of FFTs can lead to parallel scalability issues when there are many more cells along the drifting direction than in the transverse direction(s). We then describe an algorithm that has the potential to address this issue by using a higher order finite difference operator for the derivative in the plasma drifting direction, while using the standard second order operators in the transverse direction(s). The NCI for this algorithm is analyzed, and it is shown that the NCI can be eliminated using the same strategies that were used for the hybrid FFT/Finite Difference solver. This scheme also requires a current correction and filtering which require FFTs. However, we show that in this case the FFTs can be done locally on each parallel partition.

We also describe how the use of the hybrid FFT/Finite Difference or the hybrid higher order finite difference/second order finite difference methods permit combining the Lorentz boosted frame simulation technique with another ``speed up'' technique, called the quasi-3D algorithm, to gain unprecedented speed up for the LWFA simulations. In the quasi-3D algorithm the fields and currents are defined on an $r-z$ PIC grid and expanded in azimuthal harmonics. The expansion is truncated with only a few modes so it has similar computational needs of a 2D $r-z$ PIC code. We show that NCI has similar properties in $r-z$ as in $z-x$ slab geometry and show that the same strategies for eliminating the NCI in Cartesian geometry can be effective for the quasi-3D algorithm leading to the possibility of unprecedented speed up.

We also describe a new code called UPIC-EMMA that is based on fully spectral (FFT) solver. The new code includes implementation of a moving antenna that can launch lasers in the boosted frame. We also describe how the new hybrid algorithms were implemented into OSIRIS. Examples of LWFA using the boosted frame using both UPIC-EMMA and OSIRIS are given, including the comparisons against the lab frame results. We also describe how to efficiently obtain the boosted frame simulations data that are needed to generate the transformed lab frame data, as well as how to use a moving window in the boosted frame.

The NCI is also a major issue for modeling relativistic shocks with PIC algorithm. In relativistic shock simulations two counter-propagating plasmas drifting at relativistic speeds are colliding against each other. We show that the strategies for eliminating the NCI developed in this dissertation are enabling such simulations being run for much longer simulation times, which should open a path for major advances in relativistic shock research.

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