Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface PDEs) in the Time-Continuous Embedding Framework

Wai Sun Don; Jia Le Li; Leevan Ling; Bao Shan Wang; Yinghua Wang

doi:10.1007/978-3-031-76988-7_12

Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface PDEs) in the Time-Continuous Embedding Framework

Wai Sun Don
, Jia Le Li
, Leevan Ling
, Bao Shan Wang^*
, Yinghua Wang

^*Corresponding author for this work

Department of Mathematics

Research output: Chapter in book/report/conference proceeding › Conference proceeding › peer-review

1 Citation (Scopus)

Abstract

A fifth-order hybrid shock-capturing finite difference scheme (tc-Hybrid) is employed for solving one-dimensional hyperbolic conservation laws on manifolds (SPDEs). Our previously proposed time-continuous embedding approach is employed to transform SPDEs into embedded SPDEs (EPDEs) in an embedding Euclidean space [JSC 93, 84 (2022)]. The spatial gradients are discretized using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO-Z) and component-wise upwind central finite difference operators, and the ghost cell values are reconstructed using a sixth-order ENO and Lagrange interpolations in the Cartesian computational tube. The tc-Hybrid scheme identifies smooth and non-smooth regions using the robust and accurate trouble-cell detector (RBF shock-detector and Tukey’s boxplot method). This hybridization allows efficient and accurate resolution of fine-scale structures in smooth regions while capturing singular structures (shock, contact discontinuity, and rarefaction waves) in discontinuous regions in an essentially non-oscillatory manner (ENO property). The tc-Hybrid scheme has been tested on a scalar SPDE (Burgers’ equation and Buckley-Leverett problem) and a system of SPDEs (Euler equations with the Sod, Lax, and shock-density wave interaction problems) on one-dimensional manifolds with curved geometries. The results show that the tc-Hybrid scheme achieves fifth-order accuracy for smooth problems, captures singular structures in an ENO manner, and significantly reduces CPU times.

Original language	English
Title of host publication	Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023
Subtitle of host publication	Selected Papers from the ICOSAHOM Conference, Seoul, Korea, August 14 – 18, 2023
Editors	Sehun Chun, Jae-Hun Jung, Eun-Jae Park, Jie Shen
Place of Publication	Switzerland
Publisher	Springer Cham
Pages	239-255
Number of pages	17
ISBN (Electronic)	9783031769887
ISBN (Print)	9783031769870
DOIs	https://doi.org/10.1007/978-3-031-76988-7_12
Publication status	Published - 1 Nov 2025
Event	14th International Conference on Spectral and High-Order Methods - Yonsei University, Seoul, Korea, Republic of Duration: 14 Aug 2023 → 18 Aug 2023 https://link.springer.com/book/10.1007/978-3-031-76988-7 (Conference proceeding) https://icosahom2023.org/ (Conference website) https://icosahom2023.org/?page_id=90 (Conference program)

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	142
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Name	ICOSAHOM: International Conference on Spectral and High Order Methods

Conference

Conference	14th International Conference on Spectral and High-Order Methods
Abbreviated title	ICOSAHOM 2023
Country/Territory	Korea, Republic of
City	Seoul
Period	14/08/23 → 18/08/23
Internet address	https://link.springer.com/book/10.1007/978-3-031-76988-7 (Conference proceeding) https://icosahom2023.org/ (Conference website) https://icosahom2023.org/?page_id=90 (Conference program)

User-Defined Keywords

Closest point method
Euler equations
Hybrid scheme
Hyperbolic conservation laws
Manifolds
Time-continuous embedding
WENO

UN SDGs

This output contributes to the following UN Sustainable Development Goals (SDGs)

Access to Document

10.1007/978-3-031-76988-7_12

Cite this

Don, W. S., Li, J. L., Ling, L., Wang, B. S., & Wang, Y. (2025). Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface PDEs) in the Time-Continuous Embedding Framework. In S. Chun, J.-H. Jung, E.-J. Park, & J. Shen (Eds.), Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023: Selected Papers from the ICOSAHOM Conference, Seoul, Korea, August 14 – 18, 2023 (pp. 239-255). (Lecture Notes in Computational Science and Engineering; Vol. 142), (ICOSAHOM: International Conference on Spectral and High Order Methods). Springer Cham. https://doi.org/10.1007/978-3-031-76988-7_12

Don, Wai Sun ; Li, Jia Le ; Ling, Leevan et al. / Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface PDEs) in the Time-Continuous Embedding Framework. Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023: Selected Papers from the ICOSAHOM Conference, Seoul, Korea, August 14 – 18, 2023. editor / Sehun Chun ; Jae-Hun Jung ; Eun-Jae Park ; Jie Shen. Switzerland : Springer Cham, 2025. pp. 239-255 (Lecture Notes in Computational Science and Engineering). (ICOSAHOM: International Conference on Spectral and High Order Methods).

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title = "Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface PDEs) in the Time-Continuous Embedding Framework",

abstract = "A fifth-order hybrid shock-capturing finite difference scheme (tc-Hybrid) is employed for solving one-dimensional hyperbolic conservation laws on manifolds (SPDEs). Our previously proposed time-continuous embedding approach is employed to transform SPDEs into embedded SPDEs (EPDEs) in an embedding Euclidean space [JSC 93, 84 (2022)]. The spatial gradients are discretized using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO-Z) and component-wise upwind central finite difference operators, and the ghost cell values are reconstructed using a sixth-order ENO and Lagrange interpolations in the Cartesian computational tube. The tc-Hybrid scheme identifies smooth and non-smooth regions using the robust and accurate trouble-cell detector (RBF shock-detector and Tukey{\textquoteright}s boxplot method). This hybridization allows efficient and accurate resolution of fine-scale structures in smooth regions while capturing singular structures (shock, contact discontinuity, and rarefaction waves) in discontinuous regions in an essentially non-oscillatory manner (ENO property). The tc-Hybrid scheme has been tested on a scalar SPDE (Burgers{\textquoteright} equation and Buckley-Leverett problem) and a system of SPDEs (Euler equations with the Sod, Lax, and shock-density wave interaction problems) on one-dimensional manifolds with curved geometries. The results show that the tc-Hybrid scheme achieves fifth-order accuracy for smooth problems, captures singular structures in an ENO manner, and significantly reduces CPU times.",

keywords = "Closest point method, Euler equations, Hybrid scheme, Hyperbolic conservation laws, Manifolds, Time-continuous embedding, WENO",

author = "Don, \{Wai Sun\} and Li, \{Jia Le\} and Leevan Ling and Wang, \{Bao Shan\} and Yinghua Wang",

note = "Don and Ling was supported by the General Research Fund (GRF No. 12301824, 12300922) of Hong Kong Research Grant Council. Additionally, Don is partially supported by the Shandong Provincial Natural Science Foundation (ZR2022MA012). Li and B.-S. Wang are supported by the National Natural Science Foundation of China (12301530) and China Postdoctoral Science Foundation (2023M733348). Y. Wang thanks the National Natural Science Foundation of China (12401526) and the startup funding (39804138) from Nanjing Tech University. Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.; 14th International Conference on Spectral and High-Order Methods, ICOSAHOM 2023 ; Conference date: 14-08-2023 Through 18-08-2023",

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booktitle = "Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023",

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Don, WS, Li, JL, Ling, L, Wang, BS & Wang, Y 2025, Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface PDEs) in the Time-Continuous Embedding Framework. in S Chun, J-H Jung, E-J Park & J Shen (eds), Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023: Selected Papers from the ICOSAHOM Conference, Seoul, Korea, August 14 – 18, 2023. Lecture Notes in Computational Science and Engineering, vol. 142, ICOSAHOM: International Conference on Spectral and High Order Methods, Springer Cham, Switzerland, pp. 239-255, 14th International Conference on Spectral and High-Order Methods, Seoul, Korea, Republic of, 14/08/23. https://doi.org/10.1007/978-3-031-76988-7_12

Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface PDEs) in the Time-Continuous Embedding Framework. / Don, Wai Sun; Li, Jia Le; Ling, Leevan et al.
Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023: Selected Papers from the ICOSAHOM Conference, Seoul, Korea, August 14 – 18, 2023. ed. / Sehun Chun; Jae-Hun Jung; Eun-Jae Park; Jie Shen. Switzerland: Springer Cham, 2025. p. 239-255 (Lecture Notes in Computational Science and Engineering; Vol. 142), (ICOSAHOM: International Conference on Spectral and High Order Methods).

Research output: Chapter in book/report/conference proceeding › Conference proceeding › peer-review

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AU - Don, Wai Sun

AU - Li, Jia Le

AU - Ling, Leevan

AU - Wang, Bao Shan

AU - Wang, Yinghua

N1 - Don and Ling was supported by the General Research Fund (GRF No. 12301824, 12300922) of Hong Kong Research Grant Council. Additionally, Don is partially supported by the Shandong Provincial Natural Science Foundation (ZR2022MA012). Li and B.-S. Wang are supported by the National Natural Science Foundation of China (12301530) and China Postdoctoral Science Foundation (2023M733348). Y. Wang thanks the National Natural Science Foundation of China (12401526) and the startup funding (39804138) from Nanjing Tech University. Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

PY - 2025/11/1

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N2 - A fifth-order hybrid shock-capturing finite difference scheme (tc-Hybrid) is employed for solving one-dimensional hyperbolic conservation laws on manifolds (SPDEs). Our previously proposed time-continuous embedding approach is employed to transform SPDEs into embedded SPDEs (EPDEs) in an embedding Euclidean space [JSC 93, 84 (2022)]. The spatial gradients are discretized using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO-Z) and component-wise upwind central finite difference operators, and the ghost cell values are reconstructed using a sixth-order ENO and Lagrange interpolations in the Cartesian computational tube. The tc-Hybrid scheme identifies smooth and non-smooth regions using the robust and accurate trouble-cell detector (RBF shock-detector and Tukey’s boxplot method). This hybridization allows efficient and accurate resolution of fine-scale structures in smooth regions while capturing singular structures (shock, contact discontinuity, and rarefaction waves) in discontinuous regions in an essentially non-oscillatory manner (ENO property). The tc-Hybrid scheme has been tested on a scalar SPDE (Burgers’ equation and Buckley-Leverett problem) and a system of SPDEs (Euler equations with the Sod, Lax, and shock-density wave interaction problems) on one-dimensional manifolds with curved geometries. The results show that the tc-Hybrid scheme achieves fifth-order accuracy for smooth problems, captures singular structures in an ENO manner, and significantly reduces CPU times.

AB - A fifth-order hybrid shock-capturing finite difference scheme (tc-Hybrid) is employed for solving one-dimensional hyperbolic conservation laws on manifolds (SPDEs). Our previously proposed time-continuous embedding approach is employed to transform SPDEs into embedded SPDEs (EPDEs) in an embedding Euclidean space [JSC 93, 84 (2022)]. The spatial gradients are discretized using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO-Z) and component-wise upwind central finite difference operators, and the ghost cell values are reconstructed using a sixth-order ENO and Lagrange interpolations in the Cartesian computational tube. The tc-Hybrid scheme identifies smooth and non-smooth regions using the robust and accurate trouble-cell detector (RBF shock-detector and Tukey’s boxplot method). This hybridization allows efficient and accurate resolution of fine-scale structures in smooth regions while capturing singular structures (shock, contact discontinuity, and rarefaction waves) in discontinuous regions in an essentially non-oscillatory manner (ENO property). The tc-Hybrid scheme has been tested on a scalar SPDE (Burgers’ equation and Buckley-Leverett problem) and a system of SPDEs (Euler equations with the Sod, Lax, and shock-density wave interaction problems) on one-dimensional manifolds with curved geometries. The results show that the tc-Hybrid scheme achieves fifth-order accuracy for smooth problems, captures singular structures in an ENO manner, and significantly reduces CPU times.

KW - Closest point method

KW - Euler equations

KW - Hybrid scheme

KW - Hyperbolic conservation laws

KW - Manifolds

KW - Time-continuous embedding

KW - WENO

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Don WS, Li JL, Ling L, Wang BS, Wang Y. Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface PDEs) in the Time-Continuous Embedding Framework. In Chun S, Jung JH, Park EJ, Shen J, editors, Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023: Selected Papers from the ICOSAHOM Conference, Seoul, Korea, August 14 – 18, 2023. Switzerland: Springer Cham. 2025. p. 239-255. (Lecture Notes in Computational Science and Engineering). (ICOSAHOM: International Conference on Spectral and High Order Methods). doi: 10.1007/978-3-031-76988-7_12