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Volume 27, Issue 4
Troubled-Cell Indicator Based on Mean Value Property for Hybrid WENO Schemes

Kunmin Sung, Youngsoo Ha & Myungjoo Kang

Commun. Comput. Phys., 27 (2020), pp. 949-975.

Published online: 2020-02

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  • Abstract

In this paper, we introduce a new type of troubled-cell indicator to improve hybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-order linear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore, they can reduce computational cost while maintaining non-oscillatory properties in non-smooth regions. Reliable troubled-cell indicators are essential for efficient hybrid WENO methods. Most of troubled-cell indicators require proper parameters to detect discontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theorem that does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations are conducted to demonstrate the performance of the proposed indicator. The results with the proposed troubled-cell indicator are in good agreement with pure WENO schemes. Also the new indicator has advantages in the computational cost compared with the other indicators.

  • AMS Subject Headings

65M06, 65M12, 41A10, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

sgm@snu.ac.kr (Kunmin Sung)

youngamath.ha@gmail.com (Youngsoo Ha)

mkang@snu.ac.kr (Myungjoo Kang)

  • BibTex
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@Article{CiCP-27-949, author = {Sung , KunminHa , Youngsoo and Kang , Myungjoo}, title = {Troubled-Cell Indicator Based on Mean Value Property for Hybrid WENO Schemes}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {4}, pages = {949--975}, abstract = {

In this paper, we introduce a new type of troubled-cell indicator to improve hybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-order linear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore, they can reduce computational cost while maintaining non-oscillatory properties in non-smooth regions. Reliable troubled-cell indicators are essential for efficient hybrid WENO methods. Most of troubled-cell indicators require proper parameters to detect discontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theorem that does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations are conducted to demonstrate the performance of the proposed indicator. The results with the proposed troubled-cell indicator are in good agreement with pure WENO schemes. Also the new indicator has advantages in the computational cost compared with the other indicators.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0059}, url = {http://global-sci.org/intro/article_detail/cicp/14822.html} }
TY - JOUR T1 - Troubled-Cell Indicator Based on Mean Value Property for Hybrid WENO Schemes AU - Sung , Kunmin AU - Ha , Youngsoo AU - Kang , Myungjoo JO - Communications in Computational Physics VL - 4 SP - 949 EP - 975 PY - 2020 DA - 2020/02 SN - 27 DO - http://doi.org/10.4208/cicp.OA-2019-0059 UR - https://global-sci.org/intro/article_detail/cicp/14822.html KW - WENO approximation, up-wind linear approximation, troubled-cell indicator, hyperbolic conservation laws, hybrid schemes. AB -

In this paper, we introduce a new type of troubled-cell indicator to improve hybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-order linear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore, they can reduce computational cost while maintaining non-oscillatory properties in non-smooth regions. Reliable troubled-cell indicators are essential for efficient hybrid WENO methods. Most of troubled-cell indicators require proper parameters to detect discontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theorem that does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations are conducted to demonstrate the performance of the proposed indicator. The results with the proposed troubled-cell indicator are in good agreement with pure WENO schemes. Also the new indicator has advantages in the computational cost compared with the other indicators.

Kunmin Sung, Youngsoo Ha & Myungjoo Kang. (2020). Troubled-Cell Indicator Based on Mean Value Property for Hybrid WENO Schemes. Communications in Computational Physics. 27 (4). 949-975. doi:10.4208/cicp.OA-2019-0059
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