基于映射函数的中心型三阶格式  

A kind of third order central scheme based on mapping functions

作　　者：刘朋欣 李沁[1,2] 张涵信[1,2] Liu Peng xin;Li Qin;Zhang Hanxin

机构地区：[1]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,四川绵阳621000 [2]中国空气动力研究与发展中心计算空气动力研究所,四川绵阳621000

出　　处：《空气动力学学报》2017年第1期71-77,共7页Acta Aerodynamica Sinica

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)

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。High-order finite difference schemes have low numerical dissipation and dispersion error.They can capture flow structures in detail.In engineering applications,robustness be considered,while high resolution is pursued.The method of mapping function was first proposed by Henrick to adjust nonlinear weights in[0,1]for the fifth-order W EN O scheme.W ith the help of this method,the requirement of convergence order is satisfied,and the performance of the scheme is improved.Different from H enrick's method,a concept of piecewise polynomial function is proposed by authors in this paper,and the corresponding improved W ENO schemes have better resolution with the preservation of robustness.In this stu dy,afifth-orderpiecewise polynomial function(P P M5)is applied to a third order central scheme,which isreferred as the SW E N03-PP M5.H enrick's mapping function is also applied to the third order central schem e,which is referred as the SW ENO3-M.The performance of the SW ENO3-PPM5and the S W E N03-M are compared with that of the NND and the W EN()3.The analyses of approxim ate dispersion relations(ADR)indicate that the dissipation and dispersion relations of SWEN03-PPM5and SWEN03-M have an improvement compared with those of the WEN()3.The performance of the new schemes is evaluated by testing typical problems,i.e.onedimensionalShu-osher problem,two-dimensional ones including shock-boundary layer interaction problem,shock/vortex interaction problem,and hypersonic flows of the sharp double cone.The results indicate that the new schemes have better resolution than the WEN()can present a better description on complex flow structures,such as shock waves and contact discontinuity.Because the PPM5can make the final weight closer to the idea weight of the scheme,the SWEN03-PPM5possesses a little better resolution than the SWEN03-M.

关 键 词：有限差分 映射函数 中心型三阶格式 分辨率 稳定性 

分 类 号：V211.3[航空宇航科学与技术—航空宇航推进理论与工程]