Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov–Maxwell equations

Abstract

Particle-in-Cell (PIC) simulation is the most important numerical tool in plasma physics. However, its long-term accuracy has not been established. To overcome this difficulty, we developed a canonical symplectic PIC method for the Vlasov-Maxwell system by discretizing its canonical Poisson bracket. A fast local algorithm to solve the symplectic implicit time advance is discovered without root searching or global matrix inversion, enabling applications of the proposed method to very large-scale plasma simulations with many, e.g., 10 9 , degrees of freedom. The long-term accuracy and fidelity of the algorithm enables us to numerically confirm Mouhot and Villani's theory and conjecture on nonlinear Landau damping over several orders of magnitude using the PIC method, and to calculate the nonlinear evolution of the reflectivity during the mode conversion process from extraordinary waves to Bernstein waves.

Hamiltonian particle-in-cell methods for Vlasov-Maxwell equations

“…In Ref. 24, canonical symplectic methods are developed by discretising the canonical Poisson bracket directly.…”

Lorentz covariant canonical symplectic algorithms for dynamics of charged particles

“…To obtain an explicit scheme with high efficiency, here we choose the Euler-symplectic method, which can be expressed by [9,17] …”

“…It is also feasible to canonicalize the gyro-center equations to construct canonical symplectic algorithms for time-independent magnetic fields [16]. The Particle-in-Cell (PIC) method, known as the first principle simulation method for plasma systems, has been reconstructed by the use of different symplectic methods, including variational symplectic method, canonical symplectic method, and non-canonical symplectic method [6][7][8][9]. Theoretically, symplectic methods impose numerical results with a set of constrains, the number of which is determined by the freedom degrees of the systems [9,17], by preserving the global symplectic structure of the system.…”

“…The Particle-in-Cell (PIC) method, known as the first principle simulation method for plasma systems, has been reconstructed by the use of different symplectic methods, including variational symplectic method, canonical symplectic method, and non-canonical symplectic method [6][7][8][9]. Theoretically, symplectic methods impose numerical results with a set of constrains, the number of which is determined by the freedom degrees of the systems [9,17], by preserving the global symplectic structure of the system. Correspondingly, the global relative errors of motion constants can be restricted to bounded small values, which enable symplectic algorithms to retain many key properties of the origin continuous systems.…”

Explicit high-order noncanonical symplectic algorithms for ideal two-fluid systems

“…The development of geometric algorithms for these systems can be challenging. However, recently significant advances have been achieved in the development of structure preserving geometric algorithms for charged particle dynamics [37][38][39][40][41][42][43][44][45][46][47][48][49][50], the Vlasov-Maxwell systems [3,4,[51][52][53][54][55][56][57][58][59][60][61], compressible ideal MHD [62,63], and incompressible fluids [64,65]. All of these methods have demonstrated unparalleled long-term numerical accuracy and fidelity compared with conventional methods.…”