TY - JOUR T1 - A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes AU - Kang , Jian AU - Li , Xinliang JO - Communications in Computational Physics VL - 2 SP - 534 EP - 570 PY - 2020 DA - 2020/12 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2019-0112 UR - https://global-sci.org/intro/article_detail/cicp/18472.html KW - Finite difference, WENO, sufficient and necessary condition, proof. AB -

In the finite difference WENO (weighted essentially non-oscillatory) method, the final scheme on the whole stencil was constructed by linear combinations of highest order accurate schemes on sub-stencils, all of which share the same total count of grid points. The linear combination method which the original WENO applied was generalized to arbitrary positive-integer-order derivative on an arbitrary (uniform or non-uniform) mesh, still applying finite difference method. The possibility of expressing the final scheme on the whole stencil as a linear combination of highest order accurate schemes on WENO-like sub-stencils was investigated. The main results include: (a) the highest order of accuracy a finite difference scheme can achieve and (b) a sufficient and necessary condition that the linear combination exists. This is a sufficient and necessary condition for all finite difference schemes in a set (rather than a specific finite difference scheme) to have WENO-like linear combinations. After the proofs of the results, some remarks on the WENO schemes and TENO (targeted essentially non-oscillatory) schemes were given.