Explicit symplectic integrator fors-dependent static magnetic field

Variational schemes and geometric simulations for a hydrodynamic-electrodynamic model of surface plasmon polaritons

A class of variational schemes for the hydrodynamic-electrodynamic model of lossless free-electron gas in a quasineutral background is developed for high-quality simulations of surface plasmon polaritons. The Lagrangian density of lossless free-electron gas with a self-consistent electromagnetic field is established, and the dynamical equations with the associated constraints are obtained via a variational principle. Based on discrete exterior calculus, the action functional of this system is discretized and minimized to obtain the discrete dynamics. Newton-Raphson iteration and the biconjugate gradient stabilized method are equipped as a hybrid nonlinear-linear algebraic solver. Instead of discretizing the partial differential equations, the variational schemes have better numerical properties in secular simulations, as they preserve the discrete Lagrangian symplectic structure, gauge symmetry, and general energy-momentum density. Two numerical experiments were performed. The numerical results reproduce characteristic dispersion relations of bulk plasmons and surface plasmon polaritons, and the numerical errors of conserved quantities in all experiments are bounded by a small value after long-term simulations.

Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance

Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. This article proposes for arbitrary Hamiltonians similar integrators, which are explicit, of any even order, symplectic in an extended phase space, and with pleasant long time properties. They are based on a mechanical restraint that binds two copies of phase space together. Using backward error analysis, Kolmogorov-Arnold-Moser theory, and additional multiscale analysis, an error bound of O(Tδ^{l}ω) is established for integrable systems, where T,δ,l, and ω are, respectively, the (long) simulation time, step size, integrator order, and some binding constant. For nonintegrable systems with positive Lyapunov exponents, such an error bound is generally impossible, but satisfactory statistical behaviors were observed in a numerical experiment with a nonlinear Schrödinger equation.

Symplectic and energy-conserving algorithms for solving magnetic field trajectories

The exponential splitting of the classical evolution operator yields symplectic integrators if the canonical Hamiltonian is separable. Similar splitting of the noncanonical evolution operator for a charged particle in a magnetic field produces exact energy-conserving algorithms. The latter algorithms evaluate the magnetic field directly with no need of a vector potential and are more stable with far less phase errors than symplectic integrators. For a combined electric and magnetic field, these algorithms from splitting the noncanonical evolution operator are neither fully symplectic nor exactly energy conserving, yet they behave exactly like symplectic algorithms in having qualitatively correct trajectories and bounded periodic energy errors. This work shows that exponential-splitting algorithms of any order for solving particle trajectories in a general electric and magnetic field can be systematically derived by use of the angular momentum operator of quantum mechanics. The use of operator analysis in this work fully comprehends the intertwining interaction between electric and magnetic forces and makes possible the derivation of highly nontrivial integrators.