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.