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Sensitivity Parameter-Independent Characteristic-Wise Well-Balanced Finite Volume WENO Scheme for the Euler Equations Under Gravitational Fields

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Abstract

Euler equations with a gravitational source term (PDEs) admit a hydrostatic equilibrium state where the source term exactly balances the flux gradient. The property of exact preservation of the equilibria is highly desirable when the PDEs are numerically solved. Li and Xing (J Comput Phys 316:145–163, 2016) proposed a high-order well-balanced characteristic-wise finite volume weighted essentially non-oscillatory (FV-WENO) scheme for the cases of isothermal equilibrium and polytropic equilibrium. On the contrary to what was claimed, the scheme is not well-balanced. The root of the problem is the precarious effects of a non-zero sensitivity parameter in the nonlinear weights of the WENO polynomial reconstruction procedure (WENO operator). The effects are identified in the theoretical proof for the well-balanced scheme and verified numerically on a coarse mesh resolution and a long time simulation of the PDEs. In this study, two simple yet effective numerical techniques derived from the multiplicative-invariance (MI) property of a WENO operator are invoked to rectify the sensitivity parameter’s dependency yielding a correct proof for the sensitivity parameter-independent (characteristic-wise) well-balanced FV-WENO scheme. The (non-)well-balanced nature of the schemes is demonstrated with several one- and two-dimensional benchmark steady state problems and a small perturbation over the steady state problems. Moreover, the one-dimensional Sod problem under the gravitational field is also simulated for showing the performance of the well-balanced FV-WENO scheme in capturing shock, contact discontinuity, and rarefaction wave in an essentially non-oscillatory nature. It also indicates that the numerical scheme with the third-order Runge–Kutta time-stepping scheme should take the CFL number less than 0.5 to mitigate the Gibbs oscillations at the shock without increasing the numerical dissipation artificially in the Lax–Friedrichs numerical flux.

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Code Availability

The custom codes generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

We want to thank Prof. G. Li at Qingdao University and Prof. Y. Xing at Ohio State University for their support by sharing the original program used in this study and giving many valuable comments during the research. We are also grateful to the reviewers for their valuable suggestions. The research is supported by the National Natural Science Foundation of China (11801383,11871443), Hebei Provincial Natural Science Foundation (A2020210047). The author (Don) also likes to thank the Ocean University of China for providing the startup funding (201712011) to support this work.

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  1. All authors have contributed equally in this scholar work.

Authors and Affiliations

  1. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang, China

    Peng Li

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, China

    Bao-Shan Wang & Wai-Sun Don

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  1. Peng Li
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  2. Bao-Shan Wang
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Correspondence to Wai-Sun Don.

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Li, P., Wang, BS. & Don, WS. Sensitivity Parameter-Independent Characteristic-Wise Well-Balanced Finite Volume WENO Scheme for the Euler Equations Under Gravitational Fields. J Sci Comput 88, 47 (2021). https://doi.org/10.1007/s10915-021-01562-4

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  • DOI: https://doi.org/10.1007/s10915-021-01562-4

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