| Abstract: | 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. |
