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 - <p style="text-align: justify;">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.</p>