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

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,Δt)of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form (0,M_h)×(0,M_t). In other words, for each fixed value of Δt 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

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

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

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

This paper presents a new class of solvers for the subsonic compressible Navier-Stokes equations in general two- and three-dimensional multi-domains. Building up on the recent single-domain ADI-based high-order Navier-Stokes solvers (Bruno and Cubillos, Journal of Computational Physics 307 (2016) 476-495) this article presents multi-domain implicit-explicit methods of high-order of temporal accuracy. The proposed methodology incorporates: 1) A novel linear-cost implicit solver based on use of high-order backward differentiation formulae (BDF) and an alternating direction implicit approach (ADI); 2) A fast explicit solver; 3) Nearly dispersionless spectral spatial discretizations; and 4) A domain decomposition strategy that negotiates the interactions between the implicit and explicit domains. In particular, the implicit methodology is quasi-unconditionally stable (it does not suffer from CFL constraints for adequately resolved flows), and it can deliver orders of time accuracy between two and six in the presence of general boundary conditions. As demonstrated via a variety of numerical experiments in two and three dimensions, further, the proposed multi-domain parallel implicit-explicit implementations exhibit high-order convergence in space and time, robust stability properties, limited dispersion, and high parallel efficiency

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

This paper presents a new class of solvers for the subsonic compressible Navier-Stokes equations in general two- and three-dimensional multi-domains. Building up on the recent single-domain ADI-based high-order Navier-Stokes solvers (Bruno and Cubillos, Journal of Computational Physics 307 (2016) 476-495) this article presents multi-domain implicit-explicit methods of high-order of temporal accuracy. The proposed methodology incorporates: 1) A novel linear-cost implicit solver based on use of high-order backward differentiation formulae (BDF) and an alternating direction implicit approach (ADI); 2) A fast explicit solver; 3) Nearly dispersionless spectral spatial discretizations; and 4) A domain decomposition strategy that negotiates the interactions between the implicit and explicit domains. In particular, the implicit methodology is quasi-unconditionally stable (it does not suffer from CFL constraints for adequately resolved flows), and it can deliver orders of time accuracy between two and six in the presence of general boundary conditions. As demonstrated via a variety of numerical experiments in two and three dimensions, further, the proposed multi-domain parallel implicit-explicit implementations exhibit high-order convergence in space and time, robust stability properties, limited dispersion, and high parallel efficiency