Hyper Boris integrators for particle-in-cell simulation

We propose a family of numerical solvers for the nonrelativistic Newton-Lorentz equation in particle-in-cell (PIC) simulation. The new solvers extend a popular 4-step procedure, which has second-order accuracy in time, in several ways. First, we repeat the 4-step procedure n cycles, using an n-times smaller timestep (Delta_t/n). To speed up the calculation, we derive a polynomial formula for an arbitrary cycling number n, based on our earlier work (Zenitani & Kato 2020, Comput. Phys. Commun.). Second, prior to the 4-step procedure, we apply Boris-type gyrophase corrections to the electromagnetic field. In addition to the magnetic field, we amplify the electric field in an anisotropic manner to achieve higher-order (N=2,4,6... th order) accuracy. Finally, we construct a hybrid solver of the n-cycle solver and the Nth-order solver. We call it the hyper Boris solver. The (n,N) hyper Boris solver gives a numerical error of ~ (Delta_t/n)N at affordable computational cost.