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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

机译：2D和3D曲线域中可压缩的Navier-Stokes方程的高阶时间“准无条件稳定” ADI求解器

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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.
机译：本文介绍了二维和三维曲线域中可压缩的Navier-Stokes方程的时间精度较高阶的交替方向隐式（ADI）求解器（2到6阶）。时间的高阶精度源自以下因素：1）结合使用后向微分公式时间步长算法（BDF）和2）对非线性项的某些组成部分使用类似于BDF的外推技术（利用非线性解法） 3）道格拉斯-古恩（Douglas-Gunn）分裂的新颖应用（极大地简化了边界条件的处理，同时又保留了时间的高阶精度）。正如我们对各种特殊情况下算法的理论分析所建议的那样，大量的数值实验清楚地表明，本文提出的所有基于BDF的ADI算法在以下意义上都是“准无条件稳定”的：该算法对于形式为（0，M_h）×（0，M_t）的问题相关矩形邻域中的所有时空网格对（h，Δt）都是稳定的。换句话说，对于低于某个阈值的每个Δt固定值，本文介绍的Navier-Stokes求解器对于任意小的空间网格尺寸都是稳定的。对于二维空间中的线性双曲和抛物方程，二阶公式已进一步得到严格证明是无条件稳定的。尽管过去已经提出了针对名义上二阶时间精度的Navier-Stokes方程的隐式ADI解算器，但本文介绍的算法是第一个基于ADI的具有二阶或更高精度的Navier-Stokes求解器在非平凡（非周期性）边界条件下已在实践中得到验证。

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