# State of the Art Simulations of High Intense Particle Beams in Complicated Accelerator Structures

## Abstract

As a part of our research activities aimed at a detailed understanding of space charge effects in ring cyclotrons and in the corresponding injection beam lines at the Paul Scherrer Institut, we are currently developing a three dimensional space charge simulation code. Using the collision less Vlasov Maxwell equation as base model, we show how to solve this set of highly nonlinear equations in the light of accelerator simulations. We assume that we deal with low energy but high intensity beams only. Basically we use a split operator technique *H = H* _{1} + *H* _{2} enriched with the ability to impose complicated boundary conditions.

For the first term *H* _{1}, in *H* above, we use Lie Algebraic methods combined with Differential Algebraic (DA) methods to solve the single particle motion part of our problem.

Algorithms for solving kinetic equations can be roughly speaking divided into two groups, corresponding to the Lagrangian and Euler description of the phase space dynamics. We will use a parallel FFT based particle mesh approach and a Barnes Hut tree based field solver for our purposes. The efficient and accurate treatment of the space charge term *H* _{2} relies on the fact that one has a fast and accurate Poisson solver available, doing better than *O*(*n* ^{2}), where *n* characterizes the problem size. In order to minimize numerical noise, the ratio between number of macro particles *N* _{p}, and real number of particles in the beam pulse is a critical issue. With our approach we try to make this ratio close to one, in order to minimize numerical impurities. In today’s view of modern software engineering, extensibility, maintainability and re-usability are key issues, in addition to accuracy and stability. We show how to tackle the challenge of building a modern problem solving environment, by using the object oriented framework approach, hence combining two leading edge C++ class libraries, namely CLASSIC (Class Library for Acceleration System Simulation and Control) from CERN and POOMA (Parallel Object Oriented Methods and Applications) from advanced computing lab (ACL) in Los Alamos.

## Keywords

Space Charge Space Charge Effect Local Reference System Split Operator Technique Differential Algebraic## Preview

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