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A conservative discrete velocity method (DVM) is developed for theellipsoidal Fokker-Planck (ES-FP) equation in prediction of non-equilibriumneutral gas flows in this paper. The ES-FP collision operator is solved indiscrete velocity space in a concise and quick finite difference framework. Theconservation problem of discrete ES-FP collision operator is solved bymultiplying each term in it by extra conservative coefficients whose values arevery closed to unity. Their differences to unity are in the same order of thenumerical error in approximating the ES-FP operator in discrete velocity space.All the macroscopic conservative variables (mass, momentum and energy) areconserved in the present modified discrete ES-FP collision operator. Since theconservation property in discrete element of physical space is very importantfor 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 proposedfor ES-FP equation and the corresponding FP-DVM method, which can greatlyreduce the computational cost. The validity and accuracy of both ES-FP equationand FP-DVM method are examined using a series of $0D$-$3V$ homogenousrelaxation cases and $1D$-$3V$ shock structure cases with different $Mach$numbers, in which $1D$-$3V$ cases are reduced to $1D$-quasi $1V$ cases. Boththe predictions of $0D$-$3V$ and $1D$-$3V$ cases match well with the benchmarkresults such as analytical Boltzmann solution, direct full-Boltzmann numericalsolution and DSMC result. Especially, the FP-DVM predictions match well withthe DSMC results in the $Mach$ 8.0 shock structure case, which is in highnon-equilibrium, and is a challenge case of the model Boltzmann equation andthe corresponding numerical methods.

A conservative discrete velocity method for the ellipsoidal Fokker-Planck equation in gas-kinetic theory