高阶驻点上精度保持的六阶WENO有限差分格式

ORDER-PRESERVING SIXTH-ORDER WENO FINITE DIFFERENCE SCHEME IN THE PRESENCE OF HIGH ORDER CRITICAL POINTS

在五阶WENO有限差分格式的基础上,六阶WENO有限差分格式引入了额外的四点模板,减少了WENO格式的数值耗散.然而,该格式在驻点上无法达到理想收敛阶.为解决此问题,本文在非线性权重中引入整体模板的光滑性修正因子,使得驻点上非线性权重更快地收敛于理想权重,理论分析表明改进后的六阶格式能够在驻点上达到理想的六阶精度.驻点上的收敛阶测试和间断问题的数值实验表明,新提出的六阶WENO格式不仅在驻点上能够保持理想收敛精度,在间断问题上能保持本质无振荡的激波捕捉性质,同时在双曲守恒律解的光滑区域有效地求解细小尺度结构,还能够保持原有的六阶格式的计算效率.

In order to reduce the numerical dissipation of the WENO finite difference scheme,an additional four-points local substencil is introduced to construct a sixth-order centralupwind WENO finite difference scheme.However,the scheme fails to achieve the formal order of accuracy at the high order critical points.To correct this deficiency of the scheme,a smoothness modifier based on the global six-points stencil is introduced into the non-linear weights to allow a faster converge to the ideal weights at the critical points.Theoretical analysis shows that the improved sixth-order WENO scheme can achieve the ideal sixthorder accuracy at the critical points.The numerical results show that the improved scheme can maintain not only the formal order of accuracy regardless of the critical points and the essentially non-oscillatory shock-capturing property at the shocks/high gradients,while resolving fine scales structures efficiently in the smooth regions in the solution of hyperbolic conservation laws,but also the computational efficiency of the original scheme in solving the hyperbolic conservation laws.