一种三阶WENO-Z格式改进方法
An improved method for third-order WENO-Z scheme

摘要 高分辨率激波捕捉格式对含激波流场的数值模拟具有重要意义。在三阶WENO-Z格式（WENO-Z3）基础上，通过构造不同形式的全局光滑因子得到WENO-Z3N1、WENO-Z3N2、WENO-Z3N3格式。选取Sod激波管、双爆轰波碰撞、激波与熵波相互作用等经典算例，考察了3种格式（WENO-Z3N1、WENO-Z3N2、WENO-Z3N3）的计算性能。根据泰勒级数展开，理论推导给出3种格式的精度分析。通过探讨各格式理论精度与实际计算精度之间的关系得到如下结论：3种格式在连续解非极值点处的理论精度对实际计算性能起决定性的作用，并通过双马赫反射问题进一步验证该结论的可靠性。本文的研究给出一种三阶WENO-Z格式的改进方法，合理构造全局光滑因子使得格式在连续解非极值点处满足设计精度的要求。
Abstract：High resolution shock capturing schemes are of great significance for numerical simulation of flow fields containing shock waves. WENO-Z3N1, WENO-Z3N2 and WENO-Z3N3 schemes were constructed through different global smoothness indicators based on the third-order WENO-Z scheme (WENO-Z3 scheme). Several classical examples such as sod problem, interacting blast wave and shock entropy wave interaction were simulated to investigate the computing performance of the three schemes (WENO-Z3N1, WENO-Z3N2, WENO-Z3N3). Precision of the three schemes was analyzed theoretically according to Taylor series expansion. It is concluded that theoretical precision of the three schemes at non-critical points plays the important role in the actual computing performance after discussion on the relationship between the theoretical precision and actual calculation accuracy. Double Mach reflection problem was conducted to further verify the reliability of the above conclusion. The research in the present paper gives an improved method for third-order WENO-Z scheme:global smoothness indicators should be reasonably constructed to make scheme satisfy the requirement of design precision at non-critical points in the smooth flow fields.