A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes

A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes

Year:    2021

Author:    Jian Kang, Xinliang Li

Communications in Computational Physics, Vol. 29 (2021), Iss. 2 : pp. 534–570

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2019-0112

Communications in Computational Physics, Vol. 29 (2021), Iss. 2 : pp. 534–570

Published online:    2021-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    37

Keywords:    Finite difference WENO sufficient and necessary condition proof.

Author Details

Jian Kang

Xinliang Li