  We have obtained an electromagnetic PIC (EM-PIC) algorithm based on
time-space-extended particle in cell model. In this model particles are shaped
objects extended over time and space around Lagrangian markers. Sources carried
by these particles are weighted completely into centers and faces of time-space
cells of simulation-domain. Weighting method is resulted from implication of
conservation of charge of shaped particles. By solving Maxwell's equations over
source free zones of simulation grid we reduce solution of these equations to
finding field values at nods of this grid. Major source of error in this model
(and albeit other PIC models) is identified to be mismatching of particle
marker location and location of its assigned sources in time and space.
Relation of leapfrog scheme for integration of equations of motion with this
discrepancy is investigated by evaluation of violation of energy conservation.
We come in conclusion that instead of leapfrog we should integrate equations of
motion simultaneously. Though equation of particle momentum becomes time
implicit, we can solve it using a corrector-predictor method. In this way we
obtain excellent improvement in energy conservation compared to existing
leapfrog electromagnetic models. The developed theory is tested against results
of our two dimensional EM-PIC code.
