Sparse Grid Particle-in-Cell Scheme for Noise Reduction in Beam Simulations

Abstract

The complexity of standard solvers grows exponentially with the number of dimensions of the underlying equations. This issue is particularly acute for continuum solvers, which need to discretize the six-dimensional phase-space distribution function, and whose run times are consequently large even for a moderate number of grid points for each dimension. Particle-in-Cell (PIC) schemes are a popular alternate approach to continuum methods, because they only discretize the three-dimensional physical space and are therefore less subject to the curse of dimensionality. Even if so, PIC solvers still have large run times, because of the statistical error which is inherent to particle methods and which decays slowly with the number of particles per cell. In this talk, we will present a new scheme to address the curse of dimensionality and at the same time reduce the numerical noise of PIC simulations. Our PIC scheme is inspired by the sparse grids combination technique, a method invented to reduce grid based error when solving high dimensional partial differential equations [1]. The technique, when applied to the PIC method, has two benefits: 1) it almost eliminates the dependence of the grid based error on dimensionality, just like in a standard sparse grids application; 2) it lowers the statistical error significantly, because the sparse grids have larger cells, and thus a larger number of particles per cell for a given total number of particles. We will analyze the performance of our scheme for standard test problems in beam physics. We will demonstrate remarkable speed up for a certain class of problems, and less impressive performance for others. The latter will allow us to identify the limitations of our scheme and explore ideas to address them.