WENO格式权重分析与改进

Analysis and improvement of weights in WENO schemes

为了优化WENO格式计算性能,在对Jiang和Shu的经典WENO格式(记为WENO-JS)加权方法分析的基础上,通过引入间接光滑指数,构造出一种新的WENO格式——WENO-E格式,取得减小间断区耗散的效果。理论分析表明,该格式与WENO-JS格式计算效率基本相同,可达到相同阶的计算精度;但在相同网格下,较之WENO-JS格式,该格式对光滑区域的求解有更小的截断误差,对间断的捕捉有更高的分辨率。与WENO-JS格式相比,采用WENO-E格式进行线性迁移方程、非线性Burgers方程、欧拉方程等相关问题的数值实验,均能取得更好的数值结果。

In order to improve the computational performance of the WENO scheme, a new WENO scheme, namely WENO-E scheme was constructed, which reduces dissipation close to discontinuities. Based on the analysis of the algorithm for weighted factors in the classical WENO scheme （namely WENO-JS） proposed by Jiang and Shu, the new scheme was constructed by introducing indirect smooth indicator. Theoretical analysis shows that the WENO-E scheme can reach the same convergence order of WENO-JS with the same computational efficiency ; while it can obtain smaller truncation errors at smooth parts of the solution and higher resolution close to the discontinuities with the same grids than the WENO- JS. Subsequently, compared with the classical WENO scheme, when numerical experiments with the linear transport equation, the nonlinear Burgers equation and the one dimensional Euler system of equations are conducted, the WENO-E scheme achieves better numerical solutions.