  We consider a 1D-2V Vlasov-Fokker-Planck multi-species ionic description
coupled to fluid electrons. We address temporal stiffness with implicit time
stepping, suitably preconditioned. To address temperature disparity in time and
space, we extend the conservative adaptive velocity-space discretization scheme
proposed in [Taitano et al., J. Comp. Phys., 318, 391-420, (2016)] to a
spatially inhomogeneous system. In this approach, we normalize the
velocity-space coordinate to a temporally and spatially varying local
characteristic speed per species. We explicitly consider the resulting inertial
terms in the Vlasov equation, and derive a discrete formulation that conserves
mass, momentum, and energy up to a prescribed nonlinear tolerance upon
convergence. Our conservation strategy employs nonlinear constraints to enforce
these properties discretely for both the Vlasov operator and the Fokker-Planck
collision operator. Numerical examples of varying degrees of complexity,
including shock-wave propagation, demonstrate the favorable efficiency and
accuracy properties of the scheme.
