Fifth-order finite-volume weno on cylindrical grids

Mohammad Afzal Shadab; Xing Ji; Kun Xu

doi:10.1007/978-3-030-39647-3_51

Fifth-order finite-volume weno on cylindrical grids

Mohammad Afzal Shadab^*, Xing Ji, Kun Xu

^*Corresponding author for this work

Department of Mathematics

Research output: Chapter in Book/Conference Proceeding/Report › Conference Paper published in a book › peer-review

Abstract

Fifth-order finite-volume WENO is proposed for a structured grid in cylindrical coordinates. The derivation of linear weights, optimal weights uses a polynomial approximation in a dimension-by-dimension framework, implemented with the local conservation property. Finally, a Vandermonde-like system is obtained, which can be solved for linear weights on both regularly-spaced and irregularly-spaced grids in cylindrical coordinates, where the analytical relations can be derived for the former. In addition, a grid-independent formulation for evaluating the smoothness indicators is derived by minimizing the L₂-norm of the derivatives of reconstruction polynomial. The scheme converges to WENO-JS for the limiting case (R →∞). A linear stability analysis of the proposed reconstruction scheme is performed using a 1D scalar advection equation in cylindrical-radial coordinates. Several tests are performed to assess the performance of the proposed scheme. The results indicate that WENO-Curvilinear significantly improves the results when compared with the previous methods.

Original language	English
Title of host publication	Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference
Editors	Spencer J. Sherwin, Joaquim Peiró, Peter E. Vincent, David Moxey, Christoph Schwab
Publisher	Springer
Pages	637-648
Number of pages	12
ISBN (Print)	9783030396466
DOIs	https://doi.org/10.1007/978-3-030-39647-3_51
Publication status	Published - 2020
Event	12th International Conference on Spectral and High-Order Methods, ICOSAHOM 2018 - London, United Kingdom Duration: 9 Jul 2018 → 13 Jul 2018

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	134
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	12th International Conference on Spectral and High-Order Methods, ICOSAHOM 2018
Country/Territory	United Kingdom
City	London
Period	9/07/18 → 13/07/18

Bibliographical note

Publisher Copyright:
© 2020, The Author(s).

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10.1007/978-3-030-39647-3_51

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Shadab, M. A., Ji, X., & Xu, K. (2020). Fifth-order finite-volume weno on cylindrical grids. In S. J. Sherwin, J. Peiró, P. E. Vincent, D. Moxey, & C. Schwab (Eds.), Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference (pp. 637-648). (Lecture Notes in Computational Science and Engineering; Vol. 134). Springer. https://doi.org/10.1007/978-3-030-39647-3_51

Shadab, Mohammad Afzal ; Ji, Xing ; Xu, Kun. / Fifth-order finite-volume weno on cylindrical grids. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference. editor / Spencer J. Sherwin ; Joaquim Peiró ; Peter E. Vincent ; David Moxey ; Christoph Schwab. Springer, 2020. pp. 637-648 (Lecture Notes in Computational Science and Engineering).

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title = "Fifth-order finite-volume weno on cylindrical grids",

abstract = "Fifth-order finite-volume WENO is proposed for a structured grid in cylindrical coordinates. The derivation of linear weights, optimal weights uses a polynomial approximation in a dimension-by-dimension framework, implemented with the local conservation property. Finally, a Vandermonde-like system is obtained, which can be solved for linear weights on both regularly-spaced and irregularly-spaced grids in cylindrical coordinates, where the analytical relations can be derived for the former. In addition, a grid-independent formulation for evaluating the smoothness indicators is derived by minimizing the L2-norm of the derivatives of reconstruction polynomial. The scheme converges to WENO-JS for the limiting case (R →∞). A linear stability analysis of the proposed reconstruction scheme is performed using a 1D scalar advection equation in cylindrical-radial coordinates. Several tests are performed to assess the performance of the proposed scheme. The results indicate that WENO-Curvilinear significantly improves the results when compared with the previous methods.",

author = "Shadab, \{Mohammad Afzal\} and Xing Ji and Kun Xu",

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doi = "10.1007/978-3-030-39647-3\_51",

language = "English",

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Shadab, MA, Ji, X & Xu, K 2020, Fifth-order finite-volume weno on cylindrical grids. in SJ Sherwin, J Peiró, PE Vincent, D Moxey & C Schwab (eds), Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference. Lecture Notes in Computational Science and Engineering, vol. 134, Springer, pp. 637-648, 12th International Conference on Spectral and High-Order Methods, ICOSAHOM 2018, London, United Kingdom, 9/07/18. https://doi.org/10.1007/978-3-030-39647-3_51

Fifth-order finite-volume weno on cylindrical grids. / Shadab, Mohammad Afzal; Ji, Xing; Xu, Kun.
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference. ed. / Spencer J. Sherwin; Joaquim Peiró; Peter E. Vincent; David Moxey; Christoph Schwab. Springer, 2020. p. 637-648 (Lecture Notes in Computational Science and Engineering; Vol. 134).

Research output: Chapter in Book/Conference Proceeding/Report › Conference Paper published in a book › peer-review

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AU - Ji, Xing

AU - Xu, Kun

PY - 2020

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N2 - Fifth-order finite-volume WENO is proposed for a structured grid in cylindrical coordinates. The derivation of linear weights, optimal weights uses a polynomial approximation in a dimension-by-dimension framework, implemented with the local conservation property. Finally, a Vandermonde-like system is obtained, which can be solved for linear weights on both regularly-spaced and irregularly-spaced grids in cylindrical coordinates, where the analytical relations can be derived for the former. In addition, a grid-independent formulation for evaluating the smoothness indicators is derived by minimizing the L2-norm of the derivatives of reconstruction polynomial. The scheme converges to WENO-JS for the limiting case (R →∞). A linear stability analysis of the proposed reconstruction scheme is performed using a 1D scalar advection equation in cylindrical-radial coordinates. Several tests are performed to assess the performance of the proposed scheme. The results indicate that WENO-Curvilinear significantly improves the results when compared with the previous methods.

AB - Fifth-order finite-volume WENO is proposed for a structured grid in cylindrical coordinates. The derivation of linear weights, optimal weights uses a polynomial approximation in a dimension-by-dimension framework, implemented with the local conservation property. Finally, a Vandermonde-like system is obtained, which can be solved for linear weights on both regularly-spaced and irregularly-spaced grids in cylindrical coordinates, where the analytical relations can be derived for the former. In addition, a grid-independent formulation for evaluating the smoothness indicators is derived by minimizing the L2-norm of the derivatives of reconstruction polynomial. The scheme converges to WENO-JS for the limiting case (R →∞). A linear stability analysis of the proposed reconstruction scheme is performed using a 1D scalar advection equation in cylindrical-radial coordinates. Several tests are performed to assess the performance of the proposed scheme. The results indicate that WENO-Curvilinear significantly improves the results when compared with the previous methods.

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Shadab MA, Ji X, Xu K. Fifth-order finite-volume weno on cylindrical grids. In Sherwin SJ, Peiró J, Vincent PE, Moxey D, Schwab C, editors, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference. Springer. 2020. p. 637-648. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-030-39647-3_51

Fifth-order finite-volume weno on cylindrical grids

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