一种基于过积分的能量稳定通量重构方法  

AN ENERGY STABLE FLUXI RECONSTRUCTION SCHEMES BASED ONOVER-INTEGRATION

作　　者：刘冉 贾斐然 朱华君[2] 燕振国[2] 冯新龙[1] Liu Ran;Jia Feiran;Zhu Huajun;Yan Zhenguo;Feng Xinlong(Xinjiang University,School of Mathematics and Systems Science,Urumqi 830017,China;CARDC,China Aerodynamics Research and Development Center,State Key Laboratory of Aerodynamics,Mianyang 621000,China;Northwestern Polytechnical University,School of Power and Energy,Xi'an 710000,China)

机构地区：[1]新疆大学数学与系统科学学院,乌鲁木齐830017 [2]CARDC,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,绵阳621000 [3]西北工业大学动力与能源学院,西安710000

出　　处：《计算数学》2023年第3期368-384,共17页Mathematica Numerica Sinica

基　　金：国家自然科学基金(12172375,11902344);空气动力学国家重点实验室基金(SKLA 2019010101);国家数值风洞工程(NNW2019ZT4-B29)资助项目;新疆维吾尔自治区自然科学基金(2022D01D32)资助。

摘　　要：能量稳定通量重构(Energy Stable Flux Reconstruction,ESFR)方法在求解线性对流方程时具有能量稳定性质.但在求解非线性方程时能量稳定性质的实现需要采用L2投影,否则可能由于存在混淆误差,导致不稳定.本文将ESFR与过积分相结合构造具有良好去混淆效果的高阶通量重构(Flux Reconstruction,FR)方法.采用积分点大于求解点(Q> P)的取点方式,从理论上分析了格式的能量稳定特性.从数值上对比了gDG与gSD两种修正函数,三种不同过积分取点方式,并对比过积分与非过积分形式的ESFR(Q=P).通过对一维非均匀线性对流方程、二维等熵涡及欠解析涡流算例的模拟,结果表明:在gSD修正函数下,ESFR(Q> P)格式比ESFR(Q=P)格式去混淆效果更好,数值误差更小;对比两种修正函数,gDG修正函数数值误差更小,更稳定:对比三种过积分通量点分布,选定gDG修正函数时,通量点取Legendre-GaussLobatto(LGL)点或者通量点基于高斯权重剖分会具有更好的非线性稳定性,并且通量点取LGL点时最优.The Energy Stable Flux Reconstruction(ESFR)method has the property of energy stability when solving the linear convection equation.However,when solving nonlinear equations,the energy stability property requires L?projection,otherwise alising errors may lead to instability.In this paper,ESFR and over-integration are combined to construct a higher order FR scheme with good dealising effect.The energy stability of the scheme is analyzed theoretically by using the method that the integral point is larger than the solution point(Q>P).The results of using gDG and gsD correction functions and three different over-integration methods are compared numerically,and compared with ESFR(Q=P)which is not over-integration.Through the simulation of heterogeneous linear advection equation,isentropic euler vortex and under-resolved vortical fows,the results show that under the gsD correction function,the ESFR(Q>P)scheme is better than the ESFR(Q=P)scheme,and the numerical error is smaller.Compared with the two correction functions,the gDG correction function has smaller numerical error and is more stable.When the gDG correction function is selected,the flux points with Legendre-Gauss-Lobatto(LGL)points or the flux points with Gaussian weight partition have better nonlinear stability,and the flux points with LGL points are optimal.

关 键 词：高阶方法 过积分 稳定性 ESFR方法 双曲型守恒律方程 

分 类 号：O241.82[理学—计算数学]