A Spectral Canonical Electrostatic Algorithm

Studying single-particle dynamics over many periods of oscillations is a well-understood problem solved using symplectic integration. Such integration schemes derive their update sequence from an approximate Hamiltonian, guaranteeing that the geometric structure of the underlying problem is preserved. Simulating a self-consistent system over many oscillations can introduce numerical artifacts such as grid heating. This unphysical heating stems from using non-symplectic methods on Hamiltonian systems. With this guidance, we derive an electrostatic algorithm using a discrete form of Hamilton's Principle. The resulting algorithm, a gridless spectral electrostatic macroparticle model, does not exhibit the unphysical heating typical of most particle-in-cell methods. We present results of this using a two-body problem as an example of the algorithm's energy- and momentum-conserving properties.