3D Space Charge Solver Based on Tensor Decomposition for High-Intensity Beams

Abstract Accurate calculation of the space charge force inside charged particle beams is of practical concern for a deep understanding of the dynamical evolution of beams propagating in accelerators, especially for high-intensity machines. Many computational schemes have been well developed for space charge calculation. Conventional matrix-based methods are primarily suited for the 2D space charge calculation. However, one major limitation of such methods is that for the 3D case, the matrix representation becomes much more complicated. In this paper, we propose a new method via tensor representation on a discretized 3D map for space charge calculation. With the utilization of the $n$-mode product of tensors (orthogonal basis), the 3D Poisson equation can be solved in a concise manner for not only the rectangular domain, but also the nonrectangular case. It is shown that the numerical results of the new method are in good agreement with both the analytical solution of a 3D Gaussian distribution, and the numerical solution of a 2D arbitrary distribution. The paper details the mathematical derivation concerning the tensor-represented Poisson equation, the diagonalization of the Laplacian operator, and the specification of boundary conditions. The proposed method can be applied for solving 3D space charge calculations with sufficient accuracy and efficiency for large-scale 3D simulation of high-intensity beams.