Volume 29, Issue 2
A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes

Jian KangXinliang Li

Commun. Comput. Phys., 29 (2021), pp. 534-570.

Published online: 2020-12

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  • 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.

  • Keywords

Finite difference, WENO, sufficient and necessary condition, proof.

  • AMS Subject Headings

65J05, 76M20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-534, author = {Kang , Jian and Li , Xinliang}, title = {A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes}, journal = {Communications in Computational Physics}, year = {2020}, volume = {29}, number = {2}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0112}, url = {http://global-sci.org/intro/article_detail/cicp/18472.html} }
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.

Jian Kang & Xinliang Li. (2020). A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes. Communications in Computational Physics. 29 (2). 534-570. doi:10.4208/cicp.OA-2019-0112
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