  We present a new hybrid algorithm implemented in the code CHIEF (Code Hybrid
with Inertial Electron Fluid) for simulations of electron-ion plasmas. The
algorithm treats the ions kinetically, modeled by the Particle-in-Cell (PiC)
method, and electrons as an inertial fluid, modeled by electron fluid equations
without any of the approximations used in most of the other hybrid codes with
an inertial electron fluid. This kind of code is appropriate to model a large
variety of quasineutral plasma phenomena where the electron inertia and/or ion
kinetic effects are relevant. We present here the governing equations of the
model, how these are discretized and implemented numerically, as well as six
test problems to validate our numerical approach. Our chosen test problems,
where the electron inertia and ion kinetic effects play the essential role,
are: 0) Excitation of parallel eigenmodes to check numerical convergence and
stability, 1) parallel (to a background magnetic field) propagating
electromagnetic waves, 2) perpendicular propagating electrostatic waves (ion
Bernstein modes), 3) ion beam right-hand instability (resonant and
non-resonant), 4) ion Landau damping, 5) ion firehose instability, and 6) 2D
oblique ion firehose instability. Our results reproduce successfully the
predictions of linear and non-linear theory for all these problems, validating
our code. All properties of this hybrid code make it ideal to study multi-scale
phenomena between electron and ion scales such as collisionless shocks,
magnetic reconnection and kinetic plasma turbulence in the dissipation range
above the electron scales.
