Advanced 3D Poisson solvers and Particle-in-Cell methods for Accelerator Modeling
David B. Serafini
We seek to improve on the conventional FFT-based algorithms for solving the Poisson equation with infinte domain (open) boundary conditions for large problems in accelerator modeling and related areas. In particular, improvements in both accuracy and performance are possible by combining several technologies: the method of local corrections (MLC); the James algorithm; and adaptive mesh refinement (AMR).
The MLC enables the parallelization (by domain decomposition) of problems with large domains and many grid points. This improves on the FFT-based Poisson solvers typically used as it doesn't require the all-to-all communication pattern that parallel 3d FFT algorithms require, which tends to be a performance bottleneck on current (and forseeable) parallel computers. In initial tests, good scalability up to 1000 processors has been demonstrated for our new MLC solver. An essential component of our approach is a new version of the James algorithm for infinite domain boundary conditions for the case of three dimensions. By using a simplified version of the fast multipole method in the boundary-to-boundary potential calculation, we improve on the performance of the Hockney algorithm typically used by reducing the number of grid points by a factor of 8, and the CPU costs by a factor of 3. This is particularly important for large problems where computer memory limits are a consideration.
The MLC allows for the use of adaptive mesh refinement, which reduces the number of grid points and increases the accuracy in the Poisson solution. This improves on the uniform grid methods typically used in PIC codes, particularly in beam problems where the halo is large. Also, the number of particles per cell can be controlled more closely with adaptivity than with a uniform grid.
To use AMR with particles is more complicated than using uniform grids: depositing particles on the non-uniform grid; reassigning particles when the adaptive grid changes; maintaining the load balance between processors as grids and particles move. New software and algorithms are being developed to solve these problems efficiently.
We are using the Chombo AMR software framework as the basis for this work.