基于非结构四边形网格的WENO格式

Weighted Essentially Non-oscillatory Schemes on Unstructured Quadrilateral Meshes

基于非结构四边形网格发展求解双曲守恒律的三阶加权基本无振荡（WENO）格式.针对任意非结构四边形网格选取重构模板,并给出基于线性多项式的三阶线性重构.但对于一般的非结构四边形网格,会出现非常大的线性权和负权,使得非线性重构的WENO格式对光滑问题也不稳定.本文给出一个处理非常大的线性权的优化重构方法,对优化后得到的负线性权采用分裂方法进行处理.对于非线性权,提出一种考虑局部网格和物理量间断的新光滑度量因子.采用优化重构方法和新的非线性权,当前的三阶WENO格式在质量很差的网格上也具有很好的稳定性.理论的三阶精度在数值精度测试算例中得到验证,同时一范数和无穷范数的误差绝对值不依赖于网格质量;具有强间断的数值结果证明了当前格式的有效性.

A third-order weighted essentially non-oscillatory（ WENO） scheme is developed for hyperbolic conservation laws on unstructured quadrilateral meshes. As starting point of WENO reconstruction,a general stencil is proposed for any local topology on quadrilateral meshes. With selected stencil,a unified linear scheme was constructed. However,very large weights and non-negative may appear,which leads the scheme unstable even for smooth flows. An optimization approach is given to deal with very large linear weights on unstructured meshes. Splitting technique is considered to deal with negative weights obtained by optimization approach.Non-linear weight with a new smooth indicator is proposed as well. With optimization approach for very large weights and splitting technique for negative weights,the current scheme becomes more robust. Numerical tests are presented to validate accuracy. Expected convergence rate of accuracy is obtained. And absolute value of error is not affected by mesh quality. Numerical results for flow with strong discontinuities are presented to validate robustness of the WENO scheme.