Luis Chacon

An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm

Abstract This paper discusses a novel fully implicit formulation for a one-dimensional electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier implicit electrostatic PIC approaches (which are based on a linearized Vlasov–Poisson formulation), ours is based on a nonlinearly converged Vlasov–Ampere (VA) model. By iterating particles and fields to a tight nonlinear convergence tolerance, the approach features superior stability and accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit PIC implementations. In particular, the formulation is stable against temporal (Courant–Friedrichs–Lewy) and spatial (aliasing) instabilities. It is charge- and energy-conserving to numerical round-off for arbitrary implicit time steps (unlike the earlier “energy-conserving” explicit PIC formulation, which only conserves energy in the limit of arbitrarily small time steps). While momentum is not exactly conserved, errors are kept small by an adaptive particle sub-stepping orbit integrator, which is instrumental to prevent particle tunneling (a deleterious effect for long-term accuracy). The VA model is orbit-averaged along particle orbits to enforce an energy conservation theorem with particle sub-stepping. As a result, very large time steps, constrained only by the dynamical time scale of interest, are possible without accuracy loss. Algorithmically, the approach features a Jacobian-free Newton–Krylov solver. A main development in this study is the nonlinear elimination of the new-time particle variables (positions and velocities). Such nonlinear elimination, which we term particle enslavement, results in a nonlinear formulation with memory requirements comparable to those of a fluid computation, and affords us substantial freedom in regards to the particle orbit integrator. Numerical examples are presented that demonstrate the advertised properties of the scheme. In particular, long-time ion acoustic wave simulations show that numerical accuracy does not degrade even with very large implicit time steps, and that significant CPU gains are possible.

Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods

This paper explores the development of a scalable, nonlinear, fully-implicit stabilized unstructured finite element (FE) capability for 2D incompressible (reduced) resistive MHD. The discussion considers the implementation of a stabilized FE formulation in context of a fully-implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton-Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, including both low-Lundquist number (e.g., an MHD Faraday conduction pump and a hydromagnetic Rayleigh-Bernard linear stability calculation) and moderately-high Lundquist number (magnetic island coalescence problem) examples. Initial results that explore the scaling of the solution methods are presented on up to 4096 processors for problems with up to 64M unknowns on a CrayXT3/4. Additionally, a large-scale proof-of-capability calculation for 1 billion unknowns for the MHD Faraday pump problem on 24,000 cores is presented.

An efficient mixed-precision, hybrid CPU-GPU implementation of a nonlinearly implicit one-dimensional particle-in-cell algorithm

Recently, an implicit, nonlinearly consistent, energy- and charge-conserving one-dimensional (1D) particle-in-cell method has been proposed for multi-scale, full-f kinetic simulations [G. Chen et al., J. Comput. Phys. 230 (18) (2011)]. The method employs a Jacobian-free Newton-Krylov (JFNK) solver, capable of using very large timesteps without loss of numerical stability or accuracy. A fundamental feature of the method is the segregation of particle-orbit computations from the field solver, while remaining fully self-consistent. This paper describes a very efficient, mixed-precision hybrid CPU-GPU implementation of the 1D implicit PIC algorithm exploiting this feature. The JFNK solver is kept on the CPU in double precision (DP), while the implicit, charge-conserving, and adaptive particle mover is implemented on a GPU (graphics processing unit) using CUDA in single-precision (SP). Performance-oriented optimizations are introduced with the aid of the roofline model. The implicit particle mover algorithm is shown to achieve up to 400GOp/s on a Nvidia GeForce GTX580. This corresponds to 25% absolute GPU efficiency against the peak theoretical performance, and is about 100 times faster than an equivalent single-core CPU (Intel Xeon X5460) compiler-optimized execution. For the test case chosen, the mixed-precision hybrid CPU-GPU solver is shown to over-perform the DP CPU-only serial version by a factor of ~100, without apparent loss of robustness or accuracy in a challenging long-timescale ion acoustic wave simulation.

Parallel in situ indexing for data-intensive computing

As computing power increases exponentially, vast amount of data is created by many scientific research activities. However, the bandwidth for storing the data to disks and reading the data from disks has been improving at a much slower pace. These two trends produce an ever-widening data access gap. Our work brings together two distinct technologies to address this data access issue: indexing and in situ processing. From decades of database research literature, we know that indexing is an effective way to address the data access issue, particularly for accessing relatively small fraction of data records. As data sets increase in sizes, more and more analysts need to use selective data access, which makes indexing an even more important for improving data access. The challenge is that most implementations of indexing technology are embedded in large database management systems (DBMS), but most scientific datasets are not managed by any DBMS. In this work, we choose to include indexes with the scientific data instead of requiring the data to be loaded into a DBMS.We use compressed bitmap indexes from the FastBit software which are known to be highly effective for query-intensive workloads common to scientific data analysis. To use the indexes, we need to build them first. The index building procedure needs to access the whole data set and may also require a significant amount of compute time. In this work, we adapt the in situ processing technology to generate the indexes, thus removing the need of reading data from disks and to build indexes in parallel. The in situ data processing system used is ADIOS, a middleware for high-performance I/O. Our experimental results show that the indexes can improve the data access time up to 200 times depending on the fraction of data selected, and using in situ data processing system can effectively reduce the time needed to create the indexes, up to 10 times with our in situ technique when using identical parallel settings.

Short Note: A charge- and energy-conserving implicit, electrostatic particle-in-cell algorithm on mapped computational meshes

We describe the extension of the recent charge- and energy-conserving one-dimensional electrostatic particle-in-cell algorithm in Ref. [G. Chen, L. Chacon, D.C. Barnes, An energy- and charge-conserving, implicit electrostatic particle-in-cell algorithm, Journal of Computational Physics 230 (2011) 7018-7036] to mapped (body-fitted) computational meshes. The approach maintains exact charge and energy conservation properties. Key to the algorithm is a hybrid push, where particle positions are updated in logical space, while velocities are updated in physical space. The effectiveness of the approach is demonstrated with a challenging numerical test case, the ion acoustic shock wave. The generalization of the approach to multiple dimensions is outlined.

A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible (Reduced) Resistive MHD

The one-fluid visco-resistive MHD model provides a description of the dynamics of a charged fluid under the influence of an electromagnetic field. This model is strongly coupled, highly nonlinear, and characterized by physical mechanisms that span a wide range of interacting time scales. Solutions of this system can include very fast component time scales to slowly varying dynamical time scales that are long relative to the normal modes of the model equations. Fully implicit time stepping is attractive for simulating this type of wide-ranging physical phenomena. However, it is essential that one has effective preconditioning strategies so that the overall fully implicit methodology is both efficient and scalable. In this paper, we propose and explore the performance of several candidate block preconditioners for this system. One of these preconditioners is based on an operator-split approximation. This method reduces the

Development of a Consistent and Stable Fully Implicit Moment Method for Vlasov--Ampère Particle in Cell (PIC) System

3\times3

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operators...