  A customized finite-difference field solver for the particle-in-cell (PIC)
algorithm that provides higher fidelity for wave-particle interactions in
intense electromagnetic waves is presented. In many problems of interest,
particles with relativistic energies interact with intense electromagnetic
fields that have phase velocities near the speed of light. Numerical errors can
arise due to (1) dispersion errors in the phase velocity of the wave, (2) the
staggering in time between the electric and magnetic fields and between
particle velocity and position and (3) errors in the time derivative in the
momentum advance. Errors of the first two kinds are analyzed in detail. It is
shown that by using field solvers with different $\mathbf{k}$-space operators
in Faraday's and Ampere's law, the dispersion errors and magnetic field
time-staggering errors in the particle pusher can be simultaneously removed for
electromagnetic waves moving primarily in a specific direction. The new
algorithm was implemented into OSIRIS by using customized higher-order
finite-difference operators. Schemes using the proposed solver in combination
with different particle pushers are compared through PIC simulation. It is
shown that the use of the new algorithm, together with an analytic particle
pusher (assuming constant fields over a time step), can lead to accurate
modeling of the motion of a single electron in an intense laser field with
normalized vector potentials, $eA/mc^2$, exceeding $10^4$ for typical cell
sizes and time steps.
