VNL Contributions to Numerical Simulation Methods
Because the space-charge-dominated beams needed for HIFS are non-neutral plasmas, it is necessary to employ methods similar to those used in plasma physics codes to calculate dynamics. However, in contrast to the steady-state plasmas desired for MFE, the ion beams in a driver change dramatically from source to target, and their environment differs from that of stationary plasmas. Thus it has been necessary to develop methods specifically for the study of these beams. In some cases, the methods developed have broad applicability and are beginning to make their way into other applications in accelerator physics and other fields. Some spin-offs from the plasma- and beam-simulation research being carried out by the HIFS-VNL are described here.
Use of WARP by Others
The beam physics program at the University of Maryland makes extensive using of our principal beam-simulation code WARP. That group is fielding a number of small scale electron accelerator experiments involving space-charge-dominated beams, and they are using WARP both in the design of experiments and in the analysis of the results. WARP is also used by students in classes on beam physics. The VENUS project, an ECR ion source at LBNL used by the nuclear physics community, is beginning to use WARP to study the beam propagation out of the source and through the separator. This work takes advantage of WARP's flexibility in its handling of multiple species, complicated applied fields and boundary conditions, and of the bent-beam model. VENUS is a candidate source for the proposed RIA project. WARP is also being used at Michigan State University to study various high current issues in their experiments. Work has been carried out in collaboration with Dr. Kikuchi of Utsinomiya University in Japan to develop methods for initializing beams with various distributions that self-consistently include space-charge. These methods have been implemented in WARP and in a beam code being developed in Japan (this work is part of the US-Japan fusion exchange). WARP is also used in the simulation of non-neutral plasmas in traps.
The WARP code simulates beams in an electrostatic system having internal conductors by solving the Poisson equation on a 3D grid with the appropriate potentials imposed as boundary conditions. For accuracy reasons, it is insufficient simply to set the potentials at grid points equal to those of any enclosing conductors, effectively "building the conductors out of Lego bricks." Instead, the finite-difference operator in the vicinity of an internal conductor boundary is modified so that the potential interpolated along a grid line onto the conductor surface has the right value. Such methods were first developed in 3D by the HIFS group, and are propagating to other applications, most notably by their incorporation into the general-purpose package Chombo as described below, but also by the use in the radiography program of a code (GYMNOS) originally developed for HIFS.
A new and comprehensive set of models of the physics of stray electron "clouds" and gas in positive-particle accelerators is well along in development. These models account for the complex set of interactions among the various species, with the walls, and with the applied and self-fields. A "roadmap" of the required physics appears alongside. The models are beginning to be used for simulations relevant to High Energy physics accelerators, and have broad applicability to accelerators for a range of purposes. When the code has incorporated these improvements we believe that our principal beam-simulation code WARP will be without peer in the world for electron-cloud calculations. This capability is needed in many accelerators.
Outgoing-Wave Boundary Conditions
In many discrete-particle simulations of plasmas, as well as in other applications such as radar cross-section studies, the electromagnetic field components are evolved in time on spatial grids. It is highly desirable to use as compact a simulation volume as possible. However, wave reflection from the boundaries of the computational domain can lead to incorrect physics. Motivated by the need for efficient simulations of ion beam dynamics in the fusion chamber, the HIFS group developed an improved absorbing boundary condition that reflects a significantly smaller fraction of the incident wave energy than the popular Perfectly-Matched Layer method. The new method has been implemented (in 2002) in a particle-in-cell code at Ecole Polytechnique (France) that is used for fast-ignition studies, where it has become a key element that permits simulations of large domains for long times. The method has broad applicability beyond electromagnetism to such fields as acoustics and quantum mechanics).
Adaptive Mesh Refinement (AMR) with Particles
For efficiency, it is desirable to concentrate the resolution of the field grid used in particle-in-cell (PIC) calculations in those spatial regions where it is most needed. In collaboration with NERSC researchers, the HIFS group is involved in LDRD-funded research aimed at merging WARP and the AMR package Chombo. Considerable development to Chombo, including the incorporation of cut-cell boundaries and node-centered differencing, has already resulted. WARP is the pioneer application in this area, which has now broadened to include mergers of Chombo with both MFE codes and other accelerator codes.
Fundamental research into the numerical accuracy of particle-in-cell methods on non-uniform grids has been carried out. A prototype (r,z) axisymmetric AMR capability was implemented in WARP and applied to ion-diode problems. Using this method, for the first time converged time-dependent calculations are possible. In a related effort, we have learned that AMR for electromagnetic problems appears promising only when outgoing-wave boundary conditions at the coarse-fine interface are used to minimize reflections of waves which can be resolved in the coarsely zoned region. Taking advantage of the new outgoing-wave boundary condition developed by our own group (see above), a new method of correction based on the addition and subtraction of the solution on two patches at different resolutions was implemented in the Ecole Polytechnique code. Application to laser-plasma interaction simulations has provided encouraging results, and the method (or a derivative of it) will be used in 3-D production runs in the near future. Combined with the use of outgoing-wave boundary conditions to terminate the domain, this development will enable simulations on large domains over time scales that significantly exceed previous capabilities.
AMR also shows great promise in accelerator-related application. Our injector calculations have improved qualitatively in accuracy as a result of the use of such methods, and we expect that they will be widely adopted in the accelerator physics and plasma physics modeling communities.
Electromagnetic-Wave Time-Advance Methods
A related issue is the build-up of electromagnetic (EM) noise in simulations, driven by particle fluctuations. Such systems thermodynamically tend toward 1/2ĘkT per degree of freedom, and the EM field contains many degrees of freedom. The HIFS group introduced a field-advance method with a "tunable" damping rate that (relative to the mode frequency) varies as the cube of the product of the mode frequency and time step size. This method has found its way into codes used for high-power microwave tube simulation, laser-based acceleration, and fast-ignition studies. The physics of such applications has significant overlap with that of chamber propagation for heavy-ion fusion.
Nonlinear-Perturbative Beam Simulations
The PPPL HIFS group has pioneered the application of nonlinear-perturbative ("delta-f") methods to beam problems. Such methods are well suited to the study of slowly growing instabilities because they minimize discrete-particle noise. The method is being applied to accelerators for high-energy physics and other applications. HIFS work on incorporating a Darwin (magnetoinductive) field model into such calculations is relevant to MFE and other fields.
Continuum Vlasov Methods
In collaboration with colleagues in France and most recently Japan, we are exploring the application of Vlasov methods to beam problems. Such methods appear advantageous because of their large dynamic range. Since densities are computed directly rather than by counting particles in bins, regions of phase space with low density are treated with full fidelity. Such methods offer the prospect of accurate assessment of beam halo density at the wall, but are computationally expensive. However, it may suffice to apply them only over localized sections of the beam line where the parameters are changing and beam envelope "mismatch" (which can pump particles into the halo) may be expected. Thus, such a Vlasov computation may require computational resources that are reasonable in cost. HIFS work on Vlasov simulations using adaptive and non-uniform grids was viewed with interest by MFE researchers when presented to them. For accelerator systems, we invented a method for advancing the Vlasov equation for transverse beam dynamics in a quadrupole-focused system, using a modified coordinate system that moves with the mean beam motion through phase-space. This modification makes Vlasov simulation practical for such systems (work done in collaboration with E. Sonnendrucker, a visitor to the VNL).
Large-Timestep Electron Mover
The HIFS-VNL is involved directly in research employing particle "traps," specifically the Paul Trap Simulator at PPPL. In addition, the LLNL/LBNL group is working with faculty and students at UCB, to apply the HIFS discrete-particle driver simulation code WARP to pure-electron plasmas in axisymmetric traps, and to studies of quadrupole traps for the confinement of antimatter.
Other Simulation Advances
The "warped-mesh" bent-beam model which was developed for WARP (and which gave the code its name) has been adopted by Glenn Joyce and co-workers for the ELBA code and used for relativistic electron beam studies. The PPPL HIFS group played a key role in the development of a novel algorithm for solving the Darwin (magnetoinductive) field equations while avoiding the numerical instability associated with straightforward finite-differencing of the Darwin equations.