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

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• 作者：Oscar P. Brunop> ; p>p>obruno@caltech.edu" class="auth_mail" title="E-mail the corresponding authorp> ; Max Cubillos p>cubillos@caltech.edu" class="auth_mail" title="E-mail the corresponding authorp>
• 关键词：Navier&ndash ; Stokes ; Alternating direction implicit ; Quasi-unconditional stability ; High order
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：307
• 期：Complete
• 页码：476-495
• 全文大小：2352 K

文摘

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  pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999115008281&_mathId=si1.gif&_user=111111111&_pii=S0021999115008281&_rdoc=1&_issn=00219991&md5=ad7388cbdaf6ed6229311aaaec4d1c43" title="Click to view the MathML source">(h,Δt)pan>pan class="mathContainer hidden">pan class="mathCode">$(h,\mathrm{\Delta }t)$pan>pan>pan>of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form  pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999115008281&_mathId=si2.gif&_user=111111111&_pii=S0021999115008281&_rdoc=1&_issn=00219991&md5=5a1504d774f98c664e7be91ee5055be8" title="Click to view the MathML source">(0,M_h)×(0,M_t)pan>pan class="mathContainer hidden">pan class="mathCode">$(0,M_h)\times (0,M_t)$pan>pan>pan>. 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.