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

Bruno, Oscar P. and Cubillos, Max (2016) Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains. Journal of Computational Physics, 307 . pp. 476-495. ISSN 0021-9991. doi:10.1016/

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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.

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URLURL TypeDescription Paper
Bruno, Oscar P.0000-0001-8369-3014
Additional Information:© 2015 Elsevier Inc. Received 1 October 2015; Received in revised form 20 November 2015; Accepted 2 December 2015; Available online 11 December 2015. The authors gratefully acknowledge support from the Air Force Office of Scientific Research and the National Science Foundation. M.C. also thanks the National Physical Science Consortium for their support of this effort.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)UNSPECIFIED
National Physical Science Consortium (NPSC)UNSPECIFIED
Subject Keywords:Navier–Stokes; Alternating direction implicit; Quasi-unconditional stability; High order
Record Number:CaltechAUTHORS:20160218-124347606
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Official Citation:Oscar P. Bruno, Max Cubillos, Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains, Journal of Computational Physics, Volume 307, 15 February 2016, Pages 476-495, ISSN 0021-9991, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:64564
Deposited By: Tony Diaz
Deposited On:18 Feb 2016 22:37
Last Modified:10 Nov 2021 23:32

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