笛卡尔网格上可压缩Navier--Stokes方程的紧致高阶保正活性通量方法

中文摘要 AI 辅助

本文针对笛卡尔网格上的一维和二维可压缩Navier--Stokes方程，发展了一种紧致的四阶保正活性通量（AF）方法。该方法保留标准三阶AF方法的单元平均值和共享点值作为其自由度。为避免使用标准三阶AF方法的算子离散扩散项时可能出现的阶数降低，同时保持紧致性，粘性通量的散度直接采用紧致四阶算子进行离散。对于无粘部分，在有偏模板中引入下风点值可获得四阶精度。一种整体通量限制器将高阶总数值通量与低阶保正对应通量混合，联合处理无粘和粘性通量，同时保持局部守恒。结合点值的缩放限制器，该过程保证了单元平均值和点值的密度与压力正性。数值实验展示了四阶收敛性、保正性以及对激波和粘性流动结构的精确分辨。对于二维粘性激波管，与间断伽辽金方法的比较表明，该方法在分辨激波与边界层复杂相互作用时具有更高的计算效率。

英文摘要

This paper develops a compact fourth-order positivity-preserving active flux (AF) method for the one- and two-dimensional compressible Navier--Stokes equations on Cartesian meshes. The method retains the cell averages and shared point values of the standard third-order AF method as its degrees of freedom. To avoid the order reduction that can arise when diffusion is discretized using operators from the standard third-order AF method, while maintaining compactness, the divergence of the viscous flux is discretized directly using compact fourth-order operators. For the inviscid part, incorporating a downwind point value into the biased stencil yields fourth-order accuracy. A monolithic flux limiting blends high-order total numerical fluxes with low-order positivity-preserving counterparts, treating the inviscid and viscous fluxes jointly while maintaining local conservation. Together with a scaling limiter for point values, this procedure preserves density and pressure positivity for both cell averages and point values. Numerical experiments demonstrate fourth-order convergence, positivity preservation, and accurate resolution of shocks and viscous flow structures. For the two-dimensional viscous shock tube, a comparison with a discontinuous Galerkin method shows improved computational efficiency in resolving complex interaction of shock waves and boundary layers.

发表机构

• The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））
• University of Bordeaux（波尔多大学）
• University of Würzburg（维尔茨堡大学）

机构由 AI 辅助整理，请以论文原文为准。

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26 pages, 12 figures