An Adaptive, High-Order Phase-Space Remapping for the Two Dimensional Vlasov--Poisson Equations

Wang, Bei; Miller, Gregory H.; Colella, Phillip

doi:10.1137/120872954

Cited by 9 publications

(12 citation statements)

References 35 publications

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“…Such remapping or remeshing techniques have also been applied successfully in the context of fluid dynamics to vortex methods [15] and to smoothed particle hydrodynamics [16]. With remapping, PIC methods can obtain accurate numerical solutions to the Vlasov-Poisson problem for long time integrations in both the plasma [14,17] and the cosmological [18] context. However, the PIC method in [14] and [17] was only 2nd-order accurate.…”

mentioning

confidence: 99%

“…With remapping, PIC methods can obtain accurate numerical solutions to the Vlasov-Poisson problem for long time integrations in both the plasma [14,17] and the cosmological [18] context. However, the PIC method in [14] and [17] was only 2nd-order accurate. For obtaining accurate numerical solutions with a feasible number of resolution elements, higher-order methods are greatly desirable.…”

mentioning

confidence: 99%

“…In this paper, we extend the PIC algorithm of [14] and [17] to be 4th-order accurate in both space and time. The heart of our method is the use of the high-order interpolating functions derived in [20] for charge deposition and force interpolation.…”

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confidence: 99%

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A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov--Poisson Equation

Myers¹,

Colella²,

Straalen³

2017

SIAM J. Sci. Comput.

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Numerical solutions to the Vlasov-Poisson system of equations have important applications to both plasma physics and cosmology. In this paper, we present a new Particle-in-Cell (PIC) method for solving this system that is 4th-order accurate in both space and time. Our method is a high-order extension of one presented previously [B. Wang, G. Miller, and P. Colella, SIAM J. Sci. Comput., 33 (2011), pp. 3509-3537]. It treats all of the stages of the standard PIC updatecharge deposition, force interpolation, the field solve, and the particle push -with 4th-order accuracy, and includes a 6th-order accurate phase-space remapping step for controlling particle noise. We demonstrate the convergence of our method on a series of one-and two-dimensional electrostatic plasma test problems, comparing its accuracy to that of a 2nd-order method. As expected, the 4thorder method can achieve comparable accuracy to the 2nd-order method with many fewer resolution elements.

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A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov--Poisson Equation

Myers¹,

Colella²,

Straalen³

2017

SIAM J. Sci. Comput.

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“…Though this discretization will result in more grid points for the processes close to the edge, the effect from load imbalance is small considering that the number of particles is much larger than the number of grid points (e.g., particles per cell number is much greater than 1). In the case where more than 8 ghost cells are required for charge decomposition with the 4-point averaging, we can either send the off-region points explicitly to the neighboring processes [41] or completely ignore those points. For the later case, benchmarking will be required to ensure that accuracy is preserved with the approximation.…”

Section: Parallel Decompositionmentioning

confidence: 99%

Modern gyrokinetic particle-in-cell simulation of fusion plasmas on top supercomputers

Wang

Ethier

Tang

et al. 2017

The International Journal of High Performance Computing Applica

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The Gyrokinetic Toroidal Code at Princeton (GTC-P) is a highly scalable and portable particle-incell (PIC) code. It solves the 5D Vlasov-Poisson equation featuring efficient utilization of modern parallel computer architectures at the petascale and beyond. Motivated by the goal of developing a modern code capable of dealing with the physics challenge of increasing problem size with sufficient resolution, new thread-level optimizations have been introduced as well as a key additional domain decomposition. GTC-P's multiple levels of parallelism, including inter-node 2D domain decomposition and particle decomposition, as well as intra-node shared memory partition and vectorization have enabled pushing the scalability of the PIC method to extreme computational scales. In this paper, we describe the methods developed to build a highly parallelized PIC code across a broad range of supercomputer designs. This particularly includes implementations on heterogeneous systems using NVIDIA GPU accelerators and Intel Xeon Phi (MIC) co-processors and performance comparisons with state-of-the-art homogeneous HPC systems such as Blue Gene/Q. New discovery science capabilities in the magnetic fusion energy application domain are enabled, including investigations of Ion-Temperature-Gradient (ITG) driven turbulence simulations with unprecedented spatial resolution and long temporal duration. Performance studies with realistic fusion experimental parameters are carried out on multiple supercomputing systems spanning a wide range of cache capacities, cache-sharing configurations, memory bandwidth, interconnects and network topologies. These performance comparisons using a realistic discovery-sciencecapable domain application code provide valuable insights on optimization techniques across one of the broadest sets of current high-end computing platforms worldwide. arXiv:1510.05546v1 [cs.DC] 19 Oct 2015 sustainable supply of energy on Earth. Magnetic confinement fusion devices in which very high temperature plasma is confined in a tokamak, a donut-shaped vacuum vessel, is one of the most promising approaches to fusion reactors. In such a system, the interplay between the complex trajectories of individual charged particles and the collective effects arising from the long range nature of electromagnetic forces lead to a wide range of waves, oscillations, and instabilities. These include larger-scale (macro) instabilities that produce rapid topological changes in the device resulting in a catastrophic loss of fusion power, as well as smaller-scale (micro) instabilities that gradually leak energy and thus affect performance and economic viability. Turbulent fluctuations ("microturbulence") are instabilities caused by the unavoidable spatial variations (gradients) in a plasma system. They can significantly increase the transport rate of heat, particles, and momentum across the confining magnetic field and severely limit the energy confinement time for a given machine size. High-fidelity predictive numerical experiments based on 5D gyrokinet...

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“…A detailed history of methods for solving the Vlasov equation is also given in [3], whose citation list covers algorithms from early particle methods [4,5], early semi-Lagrangian methods [6], and their evolution up to the year 2011. Among the more recent developments, we point out some general trends that are common to both particle-in-cell (PIC) and mesh-based (or discrete-velocity) algorithms: energy conserving implicit formulations [7][8][9][10][11], and phase-space adaptivity [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Arbitrarily high order Convected Scheme solution of the Vlasov–Poisson system

Guclu

Christlieb

Hitchon

2014

Journal of Computational Physics

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The Convected Scheme (CS) is a 'forward-trajectory' semi-Lagrangian method for solution of transport equations, which has been most often applied to the kinetic description of plasmas and rarefied neutral gases. In its simplest form, the CS propagates the solution forward in time by advecting the so called 'moving cells' along their characteristic trajectories, and by remapping them on the mesh at the end of the time step. The CS is conservative, positivity preserving, simple to implement, and it is not subject to time step restriction to maintain stability. Recently [Y. Güçlü, W. N. G. Hitchon, A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows, J. Comput. Phys. 231 (8) (2012) 3289-3316] a new methodology was introduced for reducing numerical diffusion, based on a modified equation analysis: the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping. While the spatial accuracy was increased from 2nd to 4th order, the new scheme retained the important properties of the original method, and was shown to be extremely simple and efficient for constant advection problems.Here the CS is applied to the solution of the Vlasov-Poisson system, which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal Coulomb interactions. The Vlasov equation is split into two constant advection equations, one in configuration space and one in velocity space, and high order time accuracy is achieved by proper composition of the operators. The splitting procedure enables us to use the constant advection solver, which we extend to arbitrarily high order of accuracy in time and space: a new improved procedure is given, which makes the calculation of the corrections straightforward. Focusing on periodic domains, we describe a spectrally accurate scheme based on the fast Fourier transform; the proposed implementation is strictly conservative and positivity preserving. The ability to correctly reproduce the system dynamics, as well as resolving small-scale features in the solution, is shown in classical 1D-1V test cases, both in the linear and the non-linear regimes.

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An Adaptive, High-Order Phase-Space Remapping for the Two Dimensional Vlasov--Poisson Equations

Cited by 9 publications

References 35 publications

A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov--Poisson Equation

A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov--Poisson Equation

Modern gyrokinetic particle-in-cell simulation of fusion plasmas on top supercomputers

Arbitrarily high order Convected Scheme solution of the Vlasov–Poisson system

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