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 twoand 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 ∆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 ≤ s ≤ 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.

Higher-order implicit-explicit multi-domain compressible Navier-Stokes solvers

“…In this section we review the stability properties of the BDF-ADI methods. In [17] it was proved that the second order BDF-ADI method is unconditionally stable for linear constant coefficient advection and parabolic equations including mixed derivative terms (treated via the algorithm described in Section 4.2 above) in two dimensional space using either Fourier spectral collocation with periodic boundary conditions or Legendre collocation in a square with homogeneous Dirichlet boundary conditions. In particular, it was shown that temporal extrapolations used for the splitting and mixed derivative terms in the BDF-ADI method do not affect the unconditional stability enjoyed by the fully implicit BDF method.…”

“…The previous contributions, including [17], do not present stability proofs for the BDF-ADI methods of order higher than 2, but, in order to provide insights into the stability properties arising from the BDF time-stepping scheme in the context of time-domain PDE solvers, [17,Sec. 5.1] investigates the stability of the BDF schemes of order s ≥ 2 under periodic boundary conditions and Fourier discretizations.…”

“…One might expect that this property could be problematic for small viscosity, but Table 3 shows that BDF orders 3 and 4 are more stable than BDF orders 5 and 6 for sufficiently fine spatial discretizations. Although quasi-unconditional stability was not established for the BDF-ADI method, numerical illustrations in Section 5.2 of [17] demonstrate quantitatively, via a numerical study, that the quasi-unconditional stability property carries over to the Navier-Stokes equations as well.…”

“…Importantly, these algorithms are quasi-unconditionally stable-a concept that is defined precisely in [11], and which essentially amounts to true unconditional stability for values of Δt below a certain threshold. Reference [17] presents an extended discussion which places the observed quasi-unconditional stability of the s-order methods, with 2 ≤ s ≤ 6, on a sound theoretical basis. As demonstrated in [11], high-order temporal accuracy can be greatly advantageous in cases involving long evolution times or solutions that oscillate rapidly; methods of lower order may be more advantageous under other circumstances.…”

Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

“…(ii) They do not require use of iterative nonlinear solvers for accuracy or stability, and they rely, instead, on a BDF-like extrapolation technique [9] for certain components of the nonlinear terms; and, as established in Part II [8], (iii) They possess favorable stability properties, with rigorous unconditional-stability proofs for constant coefficient hyperbolic and parabolic equations for s = 2, and demonstrating in practice quasi-unconditional stability for 2 ≤ s ≤ 6 (Definition 1 in Section 4.2).…”