改进型加权本质无振荡格式及其对典型激波问题的高精度数值模拟

High-Accuracy Numerical Simulations on Typical Shock Problems with Improved WENO Scheme

为了提高加权本质无振荡(WENO)格式对流场结构的分辨率,提出了性能更优的WENOEX格式。在经典WENO格式的基础上,考虑各子模板光滑因子和整体模板光滑性之间的相互影响,引入表征整体模板光滑度的相关项,构造了新的光滑因子,优化了权分配。理论分析表明,WENO-EX格式保持了原有的设计精度,对间断模板的非线性权配置更大,具有更低的耗散,更高的分辨率。为验证改进格式的计算性能,采用不同的WENO格式来求解线性迁移方程控制的光滑和间断问题,并且求解了欧拉方程控制的双马赫反射和激波旋涡相互作用等几种典型激波问题。计算结果表明:WENO-EX格式在计算光滑问题时,在相同网格下可以取得更小的计算误差;在计算间断问题时,对流场结构有更高的分辨率。

In order to improve the resolution of flowing structures simulated with the weighted essentially non-oscillatory(WENO)scheme,an improved WENO scheme(WENO-EX)is constructed.By considering the interaction among the smoothness indicators of sub-stencils and whole stencil,which is based on the classic WENO scheme(WENO-JS)proposed by Jiang and Shu,a term characterizing the smoothness of the whole stencil is introduced to constitute the new smoothness indicator which changes the assignment of the nonlinear weights.The theoretical analysis shows that the improved WENO scheme can keep the original design accuracy,and obtain smaller dissipation and higher resolution with larger nonlinear weight assignment to the discontinuous stencil.Subsequently,numerical simulations for the linear transport equation with different initial conditions and typical shock problems such as the double Mach reflection and shock-vortex interaction governed by the Euler equations are conducted with different WENO schemes,and the WENO-EX scheme achieves better numerical results.The results show that WENO-EX scheme has smaller error with the same grid in smooth regions,and higher resolutionof flowing structures in dealing with problems containing discontinuities.