WENO格式权重分析与改进

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.