  A conservative discrete velocity method (DVM) is developed for the
ellipsoidal Fokker-Planck (ES-FP) equation in prediction of non-equilibrium
neutral gas flows in this paper. The ES-FP collision operator is solved in
discrete velocity space in a concise and quick finite difference framework. The
conservation problem of discrete ES-FP collision operator is solved by
multiplying each term in it by extra conservative coefficients whose values are
very closed to unity. Their differences to unity are in the same order of the
numerical error in approximating the ES-FP operator in discrete velocity space.
All the macroscopic conservative variables (mass, momentum and energy) are
conserved in the present modified discrete ES-FP collision operator. Since the
conservation property in discrete element of physical space is very important
for numerical scheme when discontinuity and large gradient exist in flow field,
a finite volume framework is adopted for the transport term of ES-FP equation.
For $nD$-$3V$ ($n<3$) cases, a $nD$-quasi $nV$ reduction is specially proposed
for ES-FP equation and the corresponding FP-DVM method, which can greatly
reduce the computational cost. The validity and accuracy of both ES-FP equation
and FP-DVM method are examined using a series of $0D$-$3V$ homogenous
relaxation cases and $1D$-$3V$ shock structure cases with different $Mach$
numbers, in which $1D$-$3V$ cases are reduced to $1D$-quasi $1V$ cases. Both
the predictions of $0D$-$3V$ and $1D$-$3V$ cases match well with the benchmark
results such as analytical Boltzmann solution, direct full-Boltzmann numerical
solution and DSMC result. Especially, the FP-DVM predictions match well with
the DSMC results in the $Mach$ 8.0 shock structure case, which is in high
non-equilibrium, and is a challenge case of the model Boltzmann equation and
the corresponding numerical methods.
