基于映射函数的中心型三阶格式

发布时间：2018-08-02 18:08

【摘要】：高阶有限差分格式具有更小的数值耗散和色散误差,可以更好地捕捉流场的精细结构。在工程计算中,追求计算格式高分辨率的同时,也要兼顾其稳定性。Henrick首先提出映射函数的方法,作用于WENO5格式的非线性权,使其满足收敛阶数要求并提高了格式的性能。不同于Henrick的构造思路,李沁等提出了一种分段多项式形式的映射函数,改进后的WENO5格式在保证稳定性的前提下,拥有更好的分辨率。本文采用映射函数的思想,将一种五次分段多项式映射函数(Piecewise Polynomial Mapping Method,PPM)运用到中心型三阶格式上,构造出SWENO3-PPM5格式;将Henrick等提出的映射函数运用到中心型三阶格式上,构造出SWENO3-M格式,并与NND、WENO3格式的性能进行比较。近似色散关系分析表明,SWENO3-PPM5和SWENO3-M的色散和耗散关系相比于WENO3得到明显的提高。典型一维算例(Shu-osher问题)、二维算例(激波/层流边界层干扰问题、激波/旋涡干扰问题、绕25°/55°尖双锥流动问题)的计算结果表明,新格式具有比NND、WENO3更高的分辨能力,可以更好地捕捉到流场中的激波、接触间断面等复杂的流动现象,且由于分段多项式映射函数相比于Henrick提出的映射函数,可以使最终的非线性权值更加接近于格式的线性权,因此SWENO3-PPM5的分辨率要稍微高于SWENO3-M。
[Abstract]:Higher order finite difference schemes have smaller numerical dissipation and dispersion errors, which can better capture the fine structure of the flow field. In engineering calculation, while pursuing the high resolution of the scheme, it is necessary to give consideration to its stability. Henrick first puts forward the method of mapping function, which acts on the nonlinear weight of the WENO5 scheme, so that it meets the requirements of convergence order and improves the performance of the scheme. Different from Henrick's construction idea, Li Qin et al proposed a piecewise polynomial mapping function. The improved WENO5 scheme has better resolution on the premise of ensuring stability. In this paper, using the idea of mapping function, we apply a quintic piecewise polynomial mapping function (Piecewise Polynomial Mapping method PPM) to the central third order format, and construct the SWENO3-PPM5 scheme, and apply the mapping function proposed by Henrick to the central third order format. The SWENO3-M format is constructed and compared with the performance of NNDN WENO3 format. The analysis of approximate dispersion relationship shows that the dispersion and dissipation relationship between SWENO3-PPM5 and SWENO3-M is obviously higher than that of WENO3. The results of typical one-dimensional numerical examples (Shu-osher problem) and two-dimensional examples (shock / laminar boundary layer interference problem, shock wave / vortex interference problem, and biconical flow around 25 掳/ 55 掳cusp) show that the new scheme has higher resolution than NND / WENO3. The complex flow phenomena such as shock waves and contact discontinuities in the flow field can be captured better. Compared with the mapping functions proposed by Henrick, the piecewise polynomial mapping functions can make the final nonlinear weights closer to the linear weights of the schemes. Therefore, the resolution of SWENO3-PPM5 is slightly higher than that of SWENO3-M.
【作者单位】：中国空气动力研究与发展中心空气动力学国家重点实验室;中国空气动力研究与发展中心计算空气动力研究所;
【基金】：国家重点基础研究发展计划(2014CB744100) 国家自然科学基金重大研究计划(91541105)
【分类号】：O241.82

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