间断Galerkin方法中的加权本质无振荡限制器述评  

A brief survey on weighted essentially non-oscillatory limiters for discontinuous Galerkin methods

作　　者：邱建贤[1] 朱君[2] QIU Jianxian;ZHU Jun(School of Mathematical Sciences,Xiamen University,Xiamen 361005,China;College of Science,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China)

机构地区：[1]厦门大学数学科学学院,福建厦门361005 [2]南京航空航天大学理学院,江苏南京211106

出　　处：《厦门大学学报（自然科学版）》2021年第3期441-452,共12页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(12071392,11872210)。

摘　　要：间断Galerkin(DG)有限元方法是当今求解可压缩双曲守恒律的一类重要的高精度数值方法,限制器是DG算法稳定的关键,用于控制DG格式在间断问题计算中产生的伪振荡进而保证格式的稳定性.针对以前存在的限制器不能保持格式精度、影响DG方法的空间紧致特性、多数不适用于多维或复杂网格体系等缺陷.本文综述了近十年来本课题组开展的一系列使用DG方法的高精度非线性限制器研究,具体包括三维非结构网格的高精度加权本质无振荡(WENO)限制器、Hermite WENO(HWENO)限制器、三角函数基空间的WENO限制器、简单紧凑型的HWENO限制器等.该系列WENO型限制器具有保持格式精度,不振荡,不含经验参数等优点,为DG方法限制器的研究开辟了一条新途径,进而丰富了该领域的基础算法研究,并具有大规模工程应用的前景.Discontinuous Galerkin(DG)method belongs to a type of important high-order numerical methods for solving compressible hyperbolic conservation laws.Limiters for DG are very important to keep the stability of the DG methods,they are used to control the spurious oscillations for solving the problems containing strong discontinuities.The weighted essentially non-oscillatory(WENO)type limiters are proposed to overcome drawbacks of some existing literature-reported limiters which cannot keep the order of accuracy and sustain the spatial compactness of DG methods.Furthermore,most of reported limiters do not suit for the multi-dimensional numerical simulations on complex meshes.In recent decades,we have studied and proposed a series of high-order nonlinear limiters,including the high-order WENO and Hermite WENO(HWENO)limiters on three-dimensional unstructured meshes,WENO limiters based on trigonometric approximation spaces,simple and compact HWENO limiters among others.These WENO limiters secure advantages of keeping the order of accuracy and suppressing spurious oscillations without introducing any empirical parameters.They open a new way of studying new limiters for DG methods and enrich the basic research in this field.Finally,these high-order DG methods with new WENO limiters also embrace a broad prospect of large scale engineering applications.

关 键 词：间断GALERKIN方法 加权本质无振荡方法 限制器 

分 类 号：O241[理学—计算数学]