Two classes of improved fifth-order weighted compact nonlinear schemes for compressible Euler equations

Huanhuan Yang; Yanqun Jiang; Qinghong Tang; Huanhuan Yang; Yanqun Jiang; Qinghong Tang

doi:10.3934/era.2025170

Electronic Research Archive

2025, Volume 33, Issue 6: 3834-3856. doi: 10.3934/era.2025170

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Two classes of improved fifth-order weighted compact nonlinear schemes for compressible Euler equations

School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang 621010, China

Received: 19 March 2025 Revised: 23 May 2025 Accepted: 04 June 2025 Published: 17 June 2025

Two enhanced fifth-order weighted compact nonlinear schemes (WCNSs), namely the WCNS scheme with the modified ZN-type weights (WCNS-ZNM) and the perturbed WCNS scheme (WCNS-P), were proposed for compressible Euler equations. The WCNS-ZNM scheme incorporates an enhanced WENO-ZN-type nonlinear interpolation technique to obtain unknown flow values at cell edges. The WCNS-P scheme introduces perturbation terms containing a free parameter into the linear interpolations, effectively reducing numerical errors in smooth regions. A monotone polynomial interpolation approach was designed to enhance numerical stability by filtering out non-smooth regions and realizing the automatic switching between the WCNS-P and WCNS-ZNM schemes in smooth and non-smooth regions. Numerical results exhibit better accuracy and performance in resolving small-scale structures and shock waves, compared to the WCNS scheme with the WENO-Z and WENO-ZN weights.
- Euler equations,
- WCNS-ZNM scheme,
- WCNS-P scheme,
- monotone polynomial interpolation,
- fifth-order accuracy
Citation: Huanhuan Yang, Yanqun Jiang, Qinghong Tang. Two classes of improved fifth-order weighted compact nonlinear schemes for compressible Euler equations[J]. Electronic Research Archive, 2025, 33(6): 3834-3856. doi: 10.3934/era.2025170

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Abstract

Two enhanced fifth-order weighted compact nonlinear schemes (WCNSs), namely the WCNS scheme with the modified ZN-type weights (WCNS-ZNM) and the perturbed WCNS scheme (WCNS-P), were proposed for compressible Euler equations. The WCNS-ZNM scheme incorporates an enhanced WENO-ZN-type nonlinear interpolation technique to obtain unknown flow values at cell edges. The WCNS-P scheme introduces perturbation terms containing a free parameter into the linear interpolations, effectively reducing numerical errors in smooth regions. A monotone polynomial interpolation approach was designed to enhance numerical stability by filtering out non-smooth regions and realizing the automatic switching between the WCNS-P and WCNS-ZNM schemes in smooth and non-smooth regions. Numerical results exhibit better accuracy and performance in resolving small-scale structures and shock waves, compared to the WCNS scheme with the WENO-Z and WENO-ZN weights.

References

[1]	X. D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200–212. https://doi.org/10.1006/jcph.1994.1187 doi: 10.1006/jcph.1994.1187
[2]	G. S. Jiang, C. W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228. https://doi.org/10.1006/jcph.1996.0130 doi: 10.1006/jcph.1996.0130
[3]	A. K. Henrick, T. D. Aslam, J. M. Powers, Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points, J. Comput. Phys., 207 (2005), 542–567. https://doi.org/10.1016/j.jcp.2005.01.023 doi: 10.1016/j.jcp.2005.01.023
[4]	R. Borges, M. Carmona, B. Costa, W. S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), 3191–3211. https://doi.org/10.1016/j.jcp.2007.11.038 doi: 10.1016/j.jcp.2007.11.038
[5]	J. Qiu, C. W. Shu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One-dimensional case, J. Comput. Phys., 193 (2004), 115–135. https://doi.org/10.1016/j.jcp.2003.07.026 doi: 10.1016/j.jcp.2003.07.026
[6]	J. Qiu, C. W. Shu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method Ⅱ: Two-dimensional case, Comput. Fluids, 34 (2005), 642–663. https://doi.org/10.1016/j.compfluid.2004.05.005 doi: 10.1016/j.compfluid.2004.05.005
[7]	J. Y. Guo, J. H. Jung, Radial basis function ENO and WENO finite difference methods based on the optimization of shape parameters, J. Sci. Comput., 70 (2017), 551–575. https://doi.org/10.1007/s10915-016-0257-y doi: 10.1007/s10915-016-0257-y
[8]	J. Y. Guo, J. H. Jung, A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method, Appl. Numer. Math., 112 (2017), 27–50. https://doi.org/10.1016/j.apnum.2016.10.003 doi: 10.1016/j.apnum.2016.10.003
[9]	B. Jeong, H. Yang, J. Yoon, Development of a WENO scheme based on radial basis function with an improved convergence order, J. Comput. Phys., 468 (2022), 111502. https://doi.org/10.1016/j.jcp.2022.111502 doi: 10.1016/j.jcp.2022.111502
[10]	T. Yang, G. Q. Zhao, Q. J. Zhao, Novel TENO schemes with improved accuracy order based on perturbed polynomial reconstruction, J. Comput. Phys., 488 (2023), 112219. https://doi.org/10.1016/j.jcp.2023.112219 doi: 10.1016/j.jcp.2023.112219
[11]	T. Yang, D. Z. Sun, Q. J. Zhao, G. Q. Zhao, X. Chen, Perturbed polynomial reconstructed seventh-order hybrid WENO scheme with improved accuracy and resolution, Comput. Fluids, 289 (2025), 106548. https://doi.org/10.1016/j.compfluid.2025.106548 doi: 10.1016/j.compfluid.2025.106548
[12]	H. M. Zuo, J. Zhu, Increasingly high-order hybrid multi-resolution WENO schemes in multi-dimensions, J. Comput. Phys., 514 (2024), 113233. https://doi.org/10.1016/j.jcp.2024.1132 33 doi: 10.1016/j.jcp.2024.113233
[13]	H. M. Zuo, J. Zhu, A new type of modified MR-WENO schemes with new troubled cell indicators for solving hyperbolic conservation laws in multi-dimensions, J. Comput. Phys., 508 (2024), 112996. https://doi.org/10.1016/j.jcp.2024.112996 doi: 10.1016/j.jcp.2024.112996
[14]	X. Y. Hu, Q. Wang, N. A. Adams, An adaptive central-upwind weighted essentially non-oscillatory scheme, J. Comput. Phys., 229 (2010), 8952–8965. https://doi.org/10.1016/j.jcp.2010.08.019 doi: 10.1016/j.jcp.2010.08.019
[15]	J. Zhu, J. X. Qiu, A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws, J. Comput. Phys., 318 (2016), 110–121. https://doi.org/10.1016/j.jcp.2016.05.010 doi: 10.1016/j.jcp.2016.05.010
[16]	Y. Shen, K. Zhang, S. Li, J. Peng, A novel method for constructing high accurate and robust WENO-Z type scheme, preprint, arXiv: 2004.07954.
[17]	S. Y. Li, Y. Q. Shen, K. Zhang, M. Yu, High order weighted essentially non-oscillatory WENO-ZN schemes for hyperbolic conservation laws, Comput. Fluids, 244 (2022), 105547. https://doi.org/10.1016/j.compfluid.2022.105547 doi: 10.1016/j.compfluid.2022.105547
[18]	X. G. Deng, H. X. Zhang, Developing high-order weighted compact nonlinear schemes, J. Comput. Phys., 165 (2000), 22–44. https://doi.org/10.1006/jcph.2000.6594 doi: 10.1006/jcph.2000.6594
[19]	T. Nonomura, K. Fujii, Robust explicit formulation of weighted compact nonlinear scheme, Comput. Fluids, 85 (2013), 8–18. https://doi.org/10.1016/j.compfluid.2012.09.001 doi: 10.1016/j.compfluid.2012.09.001
[20]	X. L. Liu, S. H. Zhang, H. X. Zhang, C. W. Shu, A new class of central compact schemes with spectral-like resolution Ⅱ: Hybrid weighted nonlinear schemes, J. Comput. Phys., 284 (2015), 133–154. https://doi.org/10.1016/j.jcp.2014.12.027 doi: 10.1016/j.jcp.2014.12.027
[21]	Z. G. Yan, H. Y. Liu, Y. K. Ma, M. L. Mao, X. G. Deng, Further improvement of weighted compact nonlinear scheme using compact nonlinear interpolation, Comput. Fluids, 156 (2017), 135–145. https://doi.org/10.1016/j.compfluid.2017.06.028 doi: 10.1016/j.compfluid.2017.06.028
[22]	M. L. Wong, S. K. Lele, High-order localized dissipation weighted compact nonlinear scheme for shock- and interface-capturing in compressible flows, J. Comput. Phys., 339 (2017), 179–209. https://doi.org/10.1016/j.jcp.2017.03.008 doi: 10.1016/j.jcp.2017.03.008
[23]	Y. Q. Jiang, S. G. Zhou, X. Zhang, Y. G. Hu, High order all-speed semi-implicit weighted compact nonlinear scheme for the isentropic Navier-Stokes equations, J. Comput. Appl. Math., 411 (2022), 114272. https://doi.org/10.1016/j.cam.2022.114272 doi: 10.1016/j.cam.2022.114272
[24]	Y. Q. Jiang, S. G. Zhou, X. Zhang, Y. G. Hu, High-order weighted compact nonlinear scheme for one- and two-dimensional Hamilton-Jacobi equations, Appl. Numer. Math., 171 (2022), 353–368. https://doi.org/10.1016/j.apnum.2021.09.012 doi: 10.1016/j.apnum.2021.09.012
[25]	X. Q. Huang, Y. Q. Jiang, H. H. Yang, Two classes of third-order weighted compact nonlinear schemes for Hamilton-Jacobi equations, Appl. Math. Comput., 469 (2024), 128554. https://doi.org/10.1016/j.amc.2024.128554 doi: 10.1016/j.amc.2024.128554
[26]	S. Pirozzoli, On the spectral properties of shock-capturing schemes, J. Comput. Phys., 219 (2006), 489–497. https://doi.org/10.1016/j.jcp.2006.07.009 doi: 10.1016/j.jcp.2006.07.009

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