3 dimensional simulations of high intense particle beams in complicated accelerator structures

Rolf Jeltsch, Andreas Adelmann

Seminar for Applied Mathematics, ETH Zürich, Switzerland

jeltsch@math.ethz.ch

http://www.sam.math.ethz.ch/ jeltsch/

Invited talk

For the first time, it is possible to do three dimensional simulations of particle beams in the complicated proton accelerator at the Paul Scherrer Institut (PSI).

After a brief introduction into accelerator theory we give the motivation for the need of 3 dimensional simulations of low energy and high intense particle beams.

The physical model is based on the collisionless Vlasov-Maxwell theory, justified by the low density ( $\approx 10^{9}$ protons/$cm^3$) of the beam and of the residual gas. The probability of large angle scattering between the protons and the residual gas is then sufficiently low, as can be estimated by considering the mean free path and the total distance a particle travels in the accelerator structure.

In this model two forces are driving the particles: the so-called external forces which originate from the exterior magnetic and electric fields which guide, focus and accelerate the beam, modeled by a relativistic Hamiltonian and internal forces arising from the Coulomb interactions. Clearly the Coulomb interactions represent a nonlinear $N^2$ problem. The number of particles $N$ is usually between $10^{9}$ and $10^{14}$.

We seperate the two forces numerically with an operator splitting. The exterior forces acting on a single particle are treated with Lie Algebraic methods combined with Taylorseries expansions. The internal Coulomb interactions are approximated by solving the Poisson equation in an open domain. We use particle-mesh method to interpolate the charge density on to a rectangular mesh. It is then Fourier transformed into Fourier space using FFT. The Hadamard-Product with the charge density and the Green's function in Fourier space are then subsequently transformed back to real space allowing us to compute efficiently the time-consuming convolution. The scalar electric field is obtained from the potential, by the use of a second-order finite difference scheme. Again by using interpolation, we find the electric field in the continuum.

The rigorous, object-oriented, parallel design and the corresponding implementation eases the extendibility and portability of the code. At present the code is available on different Linux (Beowulf) Silicon Graphics and IBM SP-2 clusters. The use of parallel Fourier transforms to solve the Poisson problem, and the full parallelisation of the split operator integration method, allows the following range of problems to be tackled: 10 to 100 million particles on meshes up to $128^2\times 2048$. The parallel efficiency is $87.5\%$ on 32 Processors; even using 128 processors we still obtain $37.5\%$ with no code optimization.

Some examples will be shown. The contents is basically the Ph.D. thesis [1] of the co-author, done at ETH and PSI.