A new central compact finite difference formula for improving robustness in weighted compact nonlinear schemes
A new central compact finite difference formula for improving robustness in weighted compact nonlinear schemes
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DOI：
10.1016/j.compfluid.2015.09.012
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发表时间：
2015-12
影响因子：
2.8
通讯作者：
T. Sumi;T. Kurotaki
中科院分区：
文献类型：
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作者：
T. Sumi;T. Kurotaki
加权紧致非线性格式（WCNS）是一种典型的有限差分方法，在求解双曲型守恒律问题时具有高精度和高分辨率。然而，当应用于严重的不连续性的流动问题，原来的WCNS往往遭受负密度或负压。我们已经发现，WCNS的鲁棒性可以显着提高，通过改变原来的中心有限差分格式中使用的WCNS的替代。本文提出了一种新的网格边和网格结点交替中心紧致格式（ACCS），并给出了一种新的WCNS（即，交替中心加权紧致非线性格式（ACWCNS）是将ACCS与非线性重构（加权插值）方法相结合而形成的。为了阐明ACWCNS的基本特性，我们进行截断误差分析，波数分析，半离散特征值分析，和一维平流方程的精度验证。随后，ACWCNS被应用到几个一维和二维基准问题的可压缩流严重的不连续性。在解决这些问题时，使用传统的WCNS很难保持密度和压力的正性;然而，ACWCNS能够提供稳定的解决方案。在光滑区域的解决方案，该计划的空间分辨率略有恶化的方法的鲁棒性提高。为了弥补这一点，在ACWCNS的非线性重建方法进行修改，并得到的计算结果与其他方法得到的结果进行了比较。
The weighted compact nonlinear scheme (WCNS) is a typical well-known finite difference method that shows high-order accuracy and high resolution when applied to problems involving hyperbolic conservation laws. However, when applied to flow problems with severe discontinuities, the original WCNS often suffers from negative densities or negative pressures. We have found that the robustness of the WCNS can be remarkably improved by changing the original central finite difference scheme employed in the WCNS to an alternative one. In this article, a new cell-edge- and cell-node-type alternate central compact scheme (ACCS) is proposed, and a new WCNS (i.e., alternate central weighted compact nonlinear scheme (ACWCNS)) is developed by combining the ACCS with a nonlinear reconstruction (weighted interpolation) method. To elucidate the basic characteristics of the ACWCNS, we perform truncation error analysis, wavenumber analysis, semi-discrete eigenvalue analysis, and accuracy validation for 1D advection equations. Subsequently, the ACWCNS is applied to several 1D and 2D benchmark problems of compressible flows with severe discontinuities. When solving such problems, it is difficult to preserve the positivity of both density and pressure using the conventional WCNS; nevertheless, the ACWCNS is able to provide stable solutions. In the smooth regions of the solutions, the spatial resolution of the scheme is found to deteriorate slightly as the robustness of the method improved. To compensate for this, the nonlinear reconstruction method in the ACWCNS is modified and the obtained computational results are compared with those obtained by alternative approaches.