On the Quasi-unconditional Stability of BDF-ADI Solvers for the Compressible Navier--Stokes Equations and Related Linear Problems

Abstract

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.