加权本质无振荡方法综述  

A brief survey on WENO methods

作　　者：邱建贤[1] 熊涛 QIU Jianxian;XIONG Tao(School of Mathematical Sciences,Xiamen University,Xiamen 361005,China)

机构地区：[1]厦门大学数学科学学院,福建厦门361005

出　　处：《厦门大学学报（自然科学版）》2023年第6期979-990,共12页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(11971025);福建省自然科学基金(2023J02003)。

摘　　要：高精度加权本质无振荡(weighted essentially non-oscillatory,WENO)格式是求解可压缩双曲守恒律的一类重要的数值格式.它基于有限差分和有限体积两类框架,通过不同模版的非线性加权组合来实现对激波等间断解的高分辨率数值模拟,并克服虚假的数值振荡.近些年来,基于非等距模板和改变加权组合方式从而提高WENO格式的鲁棒性和计算效率,高维问题结构和无结构网格的可拓展性,和对稳态问题的快速低残差收敛性仍是WENO格式设计的热门研究课题.同时将WENO格式和高阶显隐(implicit-explicit,IMEX)Runge-Kutta时间离散格式结合,应用于极端条件下的复杂流动问题的高效稳健数值模拟也是一个非常活跃的研究方向.我们开展了一系列的高精度WENO格式的设计和应用的研究,包括设计了大小非等距模板任意正线性权组合的新型WENO-ZQ格式,基于Hermite插值或重构的Hermite WENO(HWENO)格式,和对全速域欧拉、浅水波等方程组一致稳定的渐近保持WENO格式等,大大增强了WENO型格式的灵活性,也丰富了WENO格式的应用领域,将在国防工程、航空航天、天体物理、大气海洋等领域有广阔的应用前景.As an important class of numerical methods for solving compressible hyperbolic conservation laws,the high-order weighted essentially non-oscillatory(WENO)scheme belongs to both finite difference and finite volume frameworks.It achieves high resolutions of shock wave discontinuities while avoiding numerical oscillations by using a nonlinear weighted combination of different sub-stencils.In recent years,to improve the robustness and the computational efficiency,(a)the study of unequal-sized sub-stencils with a different combination,(b)a better extension to high dimensional problems with structured or non-structured meshes,and(c)smaller residue convergences for steady-state problems,have become an active research area.Furthermore,developing efficient and stable numerical methods via the combination of WENO schemes and implicit-explicit(IMEX)Runge-Kutta time discretization and then applying this combined scheme to complex flows under extreme conditions should constitute an attractive research direction.Here,we have performed a series of designs and applications for high-order WENO schemes,including(a)a new WENO-ZQ scheme with an arbitrary convex combination of unequal-sized sub-stencils,(b)Hermite WENO(HWENO)schemes with a Hermite type interpolation or reconstruction,and(c)asymptotic preserving WENO schemes with uniform stability for Euler and shallow-water equations at all speeds.These schemes have greatly improved the flexibility of WENO-type schemes and their applicable regions and can be widely applied to numerous areas,such as national defense engineering,space flight,astrophysics,atmosphere,and ocean among others.

关 键 词：加权本质无振荡方法 Hermite型加权本质无振荡方法 双曲守恒律 渐近保持 

分 类 号：O241.8[理学—计算数学]