Computational Accelerator Physics Grand Challenge: Beam Dynamics

Our goal is to develop 3D beam dynamics codes for simulating the propagation of the beam (represented as particles or as a distribution function) through an entire accelerator complex. Rather than starting from scratch we are building on the experience of Salman Habib and Robert Ryne who have been developing parallel beam dynamics codes (as well as parallel codes for astrophysical simulations and quantum mechanical simulations) at the ACL since 1993. The codes consist of 1D, 2D, and 3D Particle-in-Cell codes and Vlasov/Poisson solvers.

Particle-in-Cell Simulations

The most time consuming aspect of these simulations is the computation of the space charge potential associated with the beam. We need to treat four types of boundary conditions. The two simplest are open boundary conditions and boundary conditions that are periodic longitudinally and open transversely. These have already been implemented on the CM-5 in CM Fortran at the ACL using an approach due to Ferrell and Bertschinger. Their method is based on the use of segmented-scan routines. One of our first, and now almost completed, tasks is to implement this on the T3E at NERSC. Since domain decomposition methods implemented in message passing have been observed to have very good performance, we will also establish T3E versions of this approach due to Viktor Decyk. At the ACL, we will implement these space charge routines in Fortran90 with message passing and, to assess the usefulness of object-oriented techniques for our problems, we will also implement these using the POOMA framework. Besides the boundary conditions already mentioned, we will have to implement two more space charge routines, one to treat beams inside conducting pipes and another to treat beams in the presence of arbitrary conducting surfaces (like that found in an RFQ). These will be tied to electrostatic solvers developed in the electromagnetics portion of this project.

Direct Solvers

As mentioned above we have experience developing Vlasov/Poisson codes. These codes provide an alternative to particle methods in which the distribution function is defined on a grid in phase space. Since no particles are needed there is no sampling noise in determining the distribution function at low densities. Our codes use split-operator spectral techniques in which one alternately Fourier transforms the position-space and momentum-space of the distribution function so that certain operators can be applied to it. The main difficulty with this approach is the enormous amount of memory that is required: a 3D simulation (with a 6D phase space) that uses 64 mesh points per dimension has a total of 68 billion mesh points. Another difficulty is that the use of Fourier basis functions causes problems to develop at the boundary of the grid. An elegant solution to these problems is to use wavelet transforms instead of Fourier transforms. To be successful we need to deal with off-diagonal terms in the evolution operators associated with the wavelet bases that are not present in the Fourier method.

Fokker-Planck and Boltzmann Solvers

An area that could have important implications for beam halo formation is the presence of noise. This could be caused, for example, by vibration of beamline elements or rf noise in the power supplies. To study the effect of noise we plan a straightforward extension of our Vlasov/Poisson techniques (which model the collisionless Boltzmann equation) to treat the Fokker-Planck equation (in which collisions are described by a linear diffusion term). A more difficult problem that we will study is the development of parallel direct methods for solving the full (i.e. collisional) Boltzmann equation.

Transform-free Methods

In our previous work on parallel beam dynamics codes we used split-operator integration techniques for advancing particles (or the distribution function) in time, and during the time steps we solved Poisson's equation by convolving the appropriate Green's function with the charge density using Fast Fourier Transforms (FFTs). There is a large amount of interprocessor communication associated with FFTs, so it is worthwhile to study approaches that avoid this. We will implement an approach in which the fields on the grid are themselves advanced in time using an algorithm which explicitly satisfies the Gauss's law constraint, an algorithm due to Buneman.

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