50 research outputs found
Explicit High-Order Gauge-Independent Symplectic Algorithms for Relativistic Charged Particle Dynamics
Symplectic schemes are powerful methods for numerically integrating
Hamiltonian systems, and their long-term accuracy and fidelity have been proved
both theoretically and numerically. However direct applications of standard
symplectic schemes to relativistic charged particle dynamics result in implicit
and electromagnetic gauge-dependent algorithms. In the present study, we
develop explicit high-order gauge-independent noncanonical symplectic
algorithms for relativistic charged particle dynamics using a Hamiltonian
splitting method in the 8D phase space. It also shown that the developed
algorithms can be derived as variational integrators by appropriately
discretizing the action of the dynamics. Numerical examples are presented to
verify the excellent long-term behavior of the algorithms.Comment: 8 figure
Structure-preserving geometric particle-in-cell algorithm suppresses finite-grid instability -- Comment on "Finite grid instability and spectral fidelity of the electrostatic Particle-In-Cell algorithm'' by Huang et al
A recent paper by Huang et al. [Computer Physics Communications 207, 123
(2016)] thoroughly analyzed the Finite Grid Instability(FGI) and spectral
fidelity of standard Particle-In-Cell (PIC) methods. Numerical experiments were
carried out to demonstrate the FGIs for two PIC methods, the energy-conserving
algorithm and the momentum-conserving algorithm. The paper also suggested that
similar numerical experiments should be performed to test the newly developed
Structure-Preserving Geometric (SPG)-PIC algorithm. In this comment, we supply
the results of the suggested numerical experiments, which show that the SPG-PIC
algorithm is able to suppress the FGI
PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability
We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian
is necessarily pseudo-Hermitian regardless of whether it is diagonalizable or
not. This result is different from Mostafazadeh's, which requires the
Hamiltonian to be diagonalizable. PT-symmetry breaking often occurs at
exceptional points where the Hamiltonian is not diagonalizable. Our result
implies that PT-symmetry breaking is equivalent to the onset of instabilities
of pseudo-Hermitian systems, which was systematically studied by Krein et al.
in 1950s. In particular, we show that the mechanism of PT-symmetry breaking is
the resonance between eigenmodes with different Krein signatures.Comment: 11pages, 1 figure. arXiv admin note: text overlap with
arXiv:1801.0167
Structure-Preserving Geometric Particle-in-Cell Methods for Vlasov-Maxwell Systems
Recent development of structure-preserving geometric particle-in-cell (PIC)
algorithms for Vlasov-Maxwell systems is summarized. With the arriving of 100
petaflop and exaflop computing power, it is now possible to carry out direct
simulations of multi-scale plasma dynamics based on first-principles. However,
standard algorithms currently adopted by the plasma physics community do not
possess the long-term accuracy and fidelity required in these large-scale
simulations. This is because conventional simulation algorithms are based on
numerically solving the underpinning differential (or integro-differential)
equations, and the algorithms used in general do not preserve the geometric and
physical structures of the systems, such as the local energy-momentum
conservation law, the symplectic structure, and the gauge symmetry. As a
consequence, numerical errors accumulate coherently with time and long-term
simulation results are not reliable. To overcome this difficulty and to
hardness the power of exascale computers, a new generation of
structure-preserving geometric PIC algorithms have been developed. This new
generation of algorithms utilizes modern mathematical techniques, such as
discrete manifolds, interpolating differential forms, and non-canonical
symplectic integrators, to ensure gauge symmetry, space-time symmetry and the
conservation of charge, energy-momentum, and the symplectic structure. These
highly desired properties are difficult to achieve using the conventional PIC
algorithms. In addition to summarizing the recent development and demonstrating
practical implementations, several new results are also presented, including a
structure-preserving geometric relativistic PIC algorithm, the proof of the
correspondence between discrete gauge symmetry and discrete charge conservation
law, and a reformulation of the explicit non-canonical symplectic algorithm for
the discrete ...Comment: Submitted to Plasma Science and Technolog
What breaks parity-time-symmetry? -- pseudo-Hermiticity and resonance between positive- and negative-action modes
It is generally believed that Parity-Time (PT)-symmetry breaking occurs when
eigenvalues or both eigenvalues and eigenvectors coincide. However, we show
that this well-accepted picture of PT-symmetry breaking is incorrect. Instead,
we demonstrate that the physical mechanism of PT-symmetry breaking is the
resonance between positive- and negative-action modes. It is proved that
PT-symmetry breaking occurs when and only when this resonance condition is
satisfied, and this mechanism applies to all known PT-symmetry breakings
observed in different branches of physics. The result is achieved by proving a
remarkable fact that in finite dimensions, a PT-symmetric Hamiltonian is
necessarily pseudo-Hermitian, regardless whether it is diagonalizable or not.Comment: 15 pages, 3 figure
Field theory and structure-preserving geometric particle-in-cell algorithm for drift wave instability and turbulence
A field theory and the associated structure-preserving geometric
Particle-In-Cell (PIC) algorithm are developed to study low frequency
electrostatic perturbations with fully kinetic ions and adiabatic electrons in
magnetized plasmas. The algorithm is constructed by geometrically discretizing
the field theory using discrete exterior calculus, high-order Whitney
interpolation forms, and non-canonical Hamiltonian splitting method. The
discretization preserves the non-canonical symplectic structure of the
particle-field system, as well as the electromagnetic gauge symmetry. As a
result, the algorithm is charge-conserving and possesses long-term conservation
properties. Because drift wave turbulence and anomalous transport intrinsically
involve multi time-scales, simulation studies using fully kinetic particle
demand algorithms with long-term accuracy and fidelity. The
structure-preserving geometric PIC algorithm developed adequately servers this
purpose. The algorithm has been implemented in the \textsl{SymPIC} code, tested
and benchmarked using the examples of ion Bernstein waves and drift waves. We
apply the algorithm to study the Ion Temperature Gradient (ITG) instability and
turbulence in a 2D slab geometry. Simulation results show that at the early
stage of the turbulence, the energy diffusion is between the Bohm scaling and
gyro-Bohm scaling. At later time, the observed diffusion is closer to the
gyro-Bohm scaling, and density blobs generated by the rupture of unstable modes
are the prominent structures of the fully developed ITG turbulence
Explicit Structure-Preserving Geometric Particle-in-Cell Algorithm in Curvilinear Orthogonal Coordinate Systems and Its Applications to Whole-Device 6D Kinetic Simulations of Tokamak Physics
Explicit structure-preserving geometric Particle-in-Cell (PIC) algorithm in
curvilinear orthogonal coordinate systems is developed. The work reported
represents a further development of the structure-preserving geometric PIC
algorithm [1-12], achieving the goal of practical applications in magnetic
fusion research. The algorithm is constructed by discretizing the field theory
for the system of charged particles and electromagnetic field using Whitney
forms, discrete exterior calculus, and explicit non-canonical symplectic
integration. In addition to the truncated infinitely dimensional symplectic
structure, the algorithm preserves exactly many important physical symmetries
and conservation laws, such as local energy conservation, gauge symmetry and
the corresponding local charge conservation. As a result, the algorithm
possesses the long-term accuracy and fidelity required for
first-principles-based simulations of the multiscale tokamak physics. The
algorithm has been implemented in the SymPIC code, which is designed for
high-efficiency massively-parallel PIC simulations in modern clusters. The code
has been applied to carry out whole-device 6D kinetic simulation studies of
tokamak physics. A self-consistent kinetic steady state for fusion plasma in
the tokamak geometry is numerically found with a predominately diagonal and
anisotropic pressure tensor. The state also admits a steady-state sub-sonic ion
flow in the range of 10 km/s, agreeing with experimental observations [13, 14]
and analytical calculations [15, 16]. Kinetic ballooning instability in the
self-consistent kinetic steady state is simulated. It shows that high-n
ballooning modes have larger growth rates than low-n global modes, and in the
nonlinear phase the modes saturate approximately in 5 ion transit times at ..
Kelvin-Helmholtz instability is the result of parity-time symmetry breaking
Parity-Time (PT)-symmetry is being actively investigated as a fundamental
property of observables in quantum physics. We show that the governing
equations of the classical two-fluid interaction and the incompressible fluid
system are PT-symmetric, and the well-known Kelvin-Helmholtz instability is the
result of spontaneous PT-symmetry breaking. It is expected that all classical
conservative systems governed by Newton's law admit PT-symmetry, and the
spontaneous breaking thereof is a generic mechanism for classical
instabilities. Discovering the PT-symmetry of systems in fluid dynamics and
plasma physics and identifying the PT-symmetry breaking responsible for
instabilities enable new techniques to classical physics and enrich the physics
of PT-symmetry.Comment: 11 pages, 1 figur
Slow manifolds of classical Pauli particle enable structure-preserving geometric algorithms for guiding center dynamics
Since variational symplectic integrators for the guiding center was proposed
[1,2], structure-preserving geometric algorithms have become an active research
field in plasma physics. We found that the slow manifolds of the classical
Pauli particle enable a family of structure-preserving geometric algorithms for
guiding center dynamics with long-term stability and accuracy. This discovery
overcomes the difficulty associated with the unstable parasitic modes for
variational symplectic integrators when applied to the degenerate guiding
center Lagrangian. It is a pleasant surprise that Pauli's Hamiltonian for
electrons, which predated the Dirac equation and marks the beginning of
particle physics, reappears in classical physics as an effective algorithm for
solving an important plasma physics problem. This technique is applicable to
other degenerate Lagrangians reduced from regular Lagrangians
Simulations of relativistic-quantum plasmas using real-time lattice scalar QED
Real-time lattice quantum electrodynamics (QED) provides a unique tool for
simulating plasmas in the strong-field regime, where collective plasma scales
are not well-separated from relativistic-quantum scales. As a toy model, we
study scalar QED, which describes self-consistent interactions between charged
bosons and electromagnetic fields. To solve this model on a computer, we first
discretize the scalar-QED action on a lattice, in a way that respects geometric
structures of exterior calculus and U(1)-gauge symmetry. The lattice scalar QED
can then be solved, in the classical-statistics regime, by advancing an
ensemble of statistically equivalent initial conditions in time, using
classical field equations obtained by extremizing the discrete action. To
demonstrate the capability of our numerical scheme, we apply it to two example
problems. The first example is the propagation of linear waves, where we
recover analytic wave dispersion relations using numerical spectrum. The second
example is an intense laser interacting with a 1D plasma slab, where we
demonstrate natural transition from wakefield acceleration to pair production
when the wave amplitude exceeds the Schwinger threshold. Our real-time lattice
scheme is fully explicit and respects local conservation laws, making it
reliable for long-time dynamics. The algorithm is readily parallelized using
domain decomposition, and the ensemble may be computed using quantum
parallelism in the future.Comment: 12 pages, 7 figure