  This paper introduces alternating-direction implicit (ADI) solvers of higher
order of time-accuracy (orders two to six) for the compressible Navier-Stokes
equations in two- and three-dimensional curvilinear domains. The higher-order
accuracy in time results from 1) An application of the backward differentiation
formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like
extrapolation technique for certain components of the nonlinear terms (which
makes use of nonlinear solves unnecessary), as well as 3) A novel application
of the Douglas-Gunn splitting (which greatly facilitates handling of boundary
conditions while preserving higher-order accuracy in time). As suggested by our
theoretical analysis of the algorithms for a variety of special cases, an
extensive set of numerical experiments clearly indicate that all of the
BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally
stable" in the following sense: each algorithm is stable for all couples
$(h,\Delta t)$ of spatial and temporal mesh sizes in a problem-dependent
rectangular neighborhood of the form $(0,M_h)\times (0,M_t)$. In other words,
for each fixed value of $\Delta t$ below a certain threshold, the Navier-Stokes
solvers presented in this paper are stable for arbitrarily small spatial
mesh-sizes. The second order formulation has further been rigorously shown to
be unconditionally stable for linear hyperbolic and parabolic equations in
two-dimensional space. Although implicit ADI solvers for the Navier-Stokes
equations with nominal second-order of temporal accuracy have been proposed in
the past, the algorithms presented in this paper are the first ADI-based
Navier-Stokes solvers for which second order or better accuracy has been
verified in practice under non-trivial (non-periodic) boundary conditions.
