An Energy-Conserving Unstaggered Electromagnetic-Potential Particle-in-Cell Method, Part II: Fully Relativistic Formulation with an Adapted Higuera--Cary Pusher

@misc{christlieb2026energyconserving,
  title = {{An Energy-Conserving Unstaggered Electromagnetic-Potential Particle-in-Cell Method, Part II: Fully Relativistic Formulation with an Adapted Higuera--Cary Pusher}},
  author = {Andrew Christlieb and Luis Chacon and Sining Gong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36383},
  eprint = {2609.36383},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36383},
}

TY  - UNPB
TI  - An Energy-Conserving Unstaggered Electromagnetic-Potential Particle-in-Cell Method, Part II: Fully Relativistic Formulation with an Adapted Higuera--Cary Pusher
AU  - Andrew Christlieb
AU  - Luis Chacon
AU  - Sining Gong
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36383
UR  - https://arxiv.org/abs/2609.36383
AB  - We develop a fully relativistic extension of the energy-conserving, unstaggered electromagnetic-potential particle-in-cell method introduced in our earlier nonrelativistic work. As in the nonrelativistic generalized-momentum formulation, the fields are advanced by a Crank--Nicolson discretization of the Lorenz-gauge potential equations, the current is deposited from particle orbits, and charge is advanced through the discrete continuity equation. The new ingredient is a generalized-momentum adaptation of the Higuera--Cary relativistic particle pusher. The vector-potential force is decomposed into an orbit-discrete-gradient part, which enforces the same finite-difference chain rule used in our earlier nonrelativistic formulation, and a skew part, which generates a Higuera--Cary magnetic rotation in mechanical momentum. For prescribed fields and frozen orbit data, this generalized-momentum Higuera--Cary map is a unit-Jacobian conjugacy of the mechanical Higuera--Cary map and preserves particle phase volume in (xx,PP). For the fully coupled implicit step, the same orbit data define the current deposit, mesh-to-particle gather, vector-potential chain rule, and relativistic kinetic-energy secant velocity. The resulting particle work cancels the Crank--Nicolson field-energy balance, yielding exact relativistic total-energy conservation up to nonlinear solver tolerance, orbit-quadrature error, and roundoff. A self-adjoint Fourier projection removes modes on even-grid Nyquist planes while preserving the particle--grid work identity. Numerical tests are organized around cold relativistic two-stream and Weibel/filamentation instabilities, with diagnostics for energy drift, Gauss's law, the Lorenz gauge, particle--grid work, and the orbit chain rule.
ER  -