高效率的特征型紧致WENO混合格式

特征型紧致加权基本无振荡（WENO）混合格式HCW-R结合了迎风紧致格式CS5-P和WENO格式，具有十分优异的分辨率特性。但在求解多维方程组时，HCW-R格式需要求解块状三对角方程组，因而计算代价十分高昂。采用迎风紧致格式CS5-F代替CS5-P，构造了一个新的特征型紧致WENO混合格式HCW-E。由于HCW-E的特殊形式，其可沿迎风方向、由边界处向内推进求解，避免了处理三对角或块状三对角方程组，从而其计算代价与显式格式无异。虽然就分辨率而言，HCW-E稍逊于HCW-R，但前者的计算效率要显著高于后者。因此，当花费相同的计算代价，HCW-E格式可以获得更好的数值结果。一系列求解Euler方程组的数值试验验证了HCW-E的高分辨率特性和相比HCW-R更高的计算效率。HCW-E格式的效率优势在求解高维问题时更为明显。 

An efficient characteristic-wise hybrid compact-WENO scheme

The characteristic-wise hybrid compact-Weighted Essentially Non-Oscillatory (WENO) scheme HCW-R combines the upwind compact scheme CS5-P with the WENO scheme, achieving an excellent resolution property. However, it needs to deal with the block-tridiagonal systems of linear equations when applied to solve multi-dimensional equations. Therefore, the scheme is computationally expensive. In this paper, we construct the new characteristic-wise hybrid compact-WENO scheme HCW-E, where the upwind compact scheme CS5-F is used to replace CS5-P. Due to the special form of HCW-E, the new hybrid scheme can be solved in an advancing manner, which is along the upwind direction and from the boundary inward. In this way, the tridiagonal or block-tridiagonal systems of linear equations are avoided. As a result, the computational expense of the compact-type scheme is equivalent to that of the explicit scheme. Although the resolution of HCW-E is slightly lower than that of HCW-R, the computational efficiency of the former is significantly higher than that of the latter. Therefore, the new scheme can get better numerical results at the same cost as the previous scheme. A series of numerical experiments for solving Euler equations show that HCW-E has an excellent resolution property and is much more efficient than HCW-R. The superiority of the new scheme in computational efficiency is more significant in solving higher-dimensional problems.