Abstract
A novel method is developed to take into account realistic boundary conditions in intense nonlinear beam dynamics. The algorithm consists of three main ingredients: the boundary element method that provides a solution for the discretized reformulation of the Poisson equation as boundary integrals; a novel fast multipole method developed for accurate and efficient computation of Coulomb potentials and forces; and differential algebraic methods, which form the numerical structures that enable and hold together the different components. The fast multipole method, without any modifications, also accelerates the solution of intertwining linear systems of equations for further efficiency enhancements. The resulting algorithm scales linearly with the number of particles $N$, as $m\text{ }\mathrm{log}\text{ }m$ with the number of boundary elements $m$, and, therefore, establishes an accurate and efficient method for intense beam dynamics simulations in arbitrary enclosures. Its performance is illustrated with three different cases and structures of practical interest.
Highlights
Computational methods applied to nonlinear beam dynamics are diverse [1,2,3]
This work provides the mathematical and computational foundations for a novel fast multipole-accelerated boundary element method enabled by differential algebraic methods and optimized for nonlinear beam dynamics at the intensity frontier
The novelty of this approach comes from interweaving the fast multipole method (FMM) and the boundary element method (BEM) in the differential algebra (DA) framework and leveraging the robust flexibility this provides for the application to 3D problems
Summary
Computational methods applied to nonlinear beam dynamics are diverse [1,2,3]. The optimal choice of methods, algorithms, and codes are typically chosen based on accuracy, efficiency, complexity of the underlying problem, and computation time. This work provides the mathematical and computational foundations for a novel fast multipole-accelerated boundary element method enabled by differential algebraic methods and optimized for nonlinear beam dynamics at the intensity frontier. The novelty of this approach comes from interweaving the FMM and the BEM in the DA framework and leveraging the robust flexibility this provides for the application to 3D problems. Some elements of this work were presented in a recent dissertation [35]
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