  The companion paper "Higher-order in time quasi-unconditionally stable ADI
solvers for the compressible Navier-Stokes equations in 2D and 3D curvilinear
domains", which is referred to as Part I in what follows, introduces ADI
(Alternating Direction Implicit) solvers of higher orders of temporal accuracy
(orders $s = 2$ to $6$) for the compressible Navier-Stokes equations in two-
and three-dimensional space. The proposed methodology employs the backward
differentiation formulae (BDF) together with a quasilinear-like formulation,
high-order extrapolation for nonlinear components, and the Douglas-Gunn
splitting. A variety of numerical results presented in Part I demonstrate in
practice the theoretical convergence rates enjoyed by these algorithms, as well
as their excellent accuracy and stability properties for a wide range of
Reynolds numbers. In particular, the proposed schemes enjoy a certain property
of "quasi-unconditional stability": for small enough (problem-dependent) fixed
values of the time-step $\Delta t$, these algorithms are stable for arbitrarily
fine spatial discretizations. The present contribution presents a mathematical
basis for the performance of these algorithms. Short of providing stability
theorems for the full BDF-ADI Navier-Stokes solvers, this paper puts forth
proofs of unconditional stability and quasi-unconditional stability for BDF-ADI
schemes as well as some related un-split BDF schemes, for a variety of related
linear model problems in one, two and three spatial dimensions, and for schemes
of orders $2\leq s\leq 6$ of temporal accuracy. Additionally, a set of
numerical tests presented in this paper for the compressible Navier-Stokes
equation indicate that quasi-unconditional stability carries over to the fully
non-linear context.
