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 timestep Δt, these algorithms are stable for arbitrarily fine spatial discretizations. The present contribution presents a mathematical basis for the observed performance of these algorithms. Short of providing stability theorems for the full Navier-Stokes BDF-ADI solvers, this paper puts forth a number of stability proofs for BDF-ADI schemes as well as some related unsplit BDF schemes for a variety of related linear model problems in one, two, and three spatial dimensions. These include proofs of quasi-unconditional stability for unsplit BDF schemes of orders 2 ≤ s ≤ 6, and even a proof of a form of unconditional stability for two-dimensional BDF-ADI schemes of order 2 for both convection and diffusion problems. 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 nonlinear context.

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