  The use of explicit particle-in-cell (PIC) method for relativistic plasma
simulations is restricted by numerical heating and instabilities that may
significantly constrain the choice of time and space steps. To partially
eliminate these limitations we consider a possibility to enforce exact energy
conservation by altering the standard time step splitting. Instead of updating
particles in a given field and then the field using the current they produce,
we consider subsystems that describe the coupling of a single particle and the
field at the nearby nodes and solve them with enforced energy conservation
sequentially for all particles, which is completed by the field update with
zero current. Such an approach is compatible with various advances, ranging
from accounting for additional physical effects to the use of
numerical-dispersion-free field solvers, high-order weighting shapes and
particle push subcycling. To facilitate further considerations and use, we
provide a basic implementation in a 3D, relativistic, spectral code $\pi$-PIC,
which we make publicly available. The method and its implementations are
verified using simulations of cold plasma oscillations, Landau damping and
relativistic two-stream instability. The capabilities of the method to deal
with large time and space steps are demonstrated in the problem of plasma
heating by intense incident radiation.
