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基于边界变差最小的高精度有限差分格式构造-2021增刊

张昊1,谢春晖1,董义道2,王东方3,邓小刚2   

  1. 1. 国防科技大学
    2.
    3. 国防科技大学航天科学与工程学院
  • 收稿日期:2021-09-18 修回日期:2021-10-19 出版日期:2021-10-21 发布日期:2021-10-21
  • 通讯作者: 谢春晖

Constructing high-order finite difference scheme based on boundary variation diminishing principle

  • Received:2021-09-18 Revised:2021-10-19 Online:2021-10-21 Published:2021-10-21

摘要: 从加权紧致非线性(WCNS)类高精度有限差分格式出发,在其重构形式的基础上,根据边界变差最小(BVD)原理,遵循单元边界两侧重构物理量值之差最小的准则,在每个单元内通过两步空间重构,构造了一种新的高精度有限差分格式。一般对WCNS等加权非线性格式的改进都是基于改善色散耗散特性、优化非线性权、提高分辨率等单一途径,本文将它们作为重构候补函数进行结合,既保持了各自优势所在,又能控制格式整体粘性,所得格式具有丰富的应用场景。通过数值实验,将结果与单一格式进行对比,新格式既能在流场光滑区保证设计的精度,对激波等间断附近的振荡也有很好的抑制作用,提高了对高波数区的分辨率,而且在长时间计算后也有较为精确的结果。面向广泛发展的数值格式,还可以构造出其他新方法,对包含强间断和多尺度的流动问题可以获得更好的结果。

关键词: 有限差分离散, 高精度格式, WCNS, 激波捕捉, 空间重构

Abstract: A novel high-order finite difference scheme is proposed originating from Weighted Compact Nonlinear Scheme (WCNS). On the basis of WCNS, a two-stage spatial reconstruction is implemented in each cell following Boundary Variation Di-minishing (BVD) principle, which requires the difference between reconstructed values of physical quantity at cell bound-aries to be minimum. Generally, improvements on weighted nonlinear schemes are in view of certain single method, such as enhancing dispersion and diffusion properties, improving nonlinear weights, and increasing resolution. Here they are in combination as candidate functions for reconstruction. The new scheme can not only retain each candidate’s own ad-vantage, but also control viscosity of the whole scheme, therefore possessing various kinds of application scenarios. Nu-merical experiments, including accuracy tests, shock tube problems and double Mach reflection, are conducted from present scheme to other single ones. Results reveal that the new scheme is capable to attain designed accuracy in smooth area of flow field, and decrease spurious oscillations near shocks, which increases resolution on high wave num-ber regions. In addition, precise wave profile can be acquired even after long-time simulation. With the help of extensively developed numerical schemes, other new approaches could be formulated for compressible flows to provide better prop-erties on strong discontinuities and multi-scale structures.

Key words: finite difference discretization, high-order accurate scheme, WCNS, shock capturing, space reconstruction

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