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Well-Balanced Discontinuous Galerkin Methods for the Euler Equations Under Gravitational Fields

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Abstract

Euler equations under gravitational field admit hydrostatic equilibrium state where the flux produced by the pressure is exactly balanced by the gravitational source term. In this paper, we present well-balanced Runge–Kutta discontinuous Galerkin methods which can preserve the isothermal hydrostatic balance state exactly and maintain genuine high order accuracy for general solutions. To obtain the well-balanced property, we first reformulate the source term, and then approximate it in a way which mimics the discontinuous Galerkin approximation of the flux term. Extensive one- and two-dimensional simulations are performed to verify the properties of these schemes such as the exact preservation of the hydrostatic balance state, the ability to capture small perturbation of such state, and the genuine high order accuracy in smooth regions.

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Acknowledgments

The research of the first author is supported by the National Natural Science Foundation of P.R. China (No. 11201254, 11401332) and the Project for Scientific Plan of Higher Education in Shandong Providence of P.R. China (No. J12LI08). This work was partially performed at the State Key Laboratory of Science/Engineering Computing of P.R. China by virtue of the computational resources of Professor Li Yuan’s group. The first author is also thankful to Professor Li Yuan for his kind invitation. The research of the second author is sponsored by NSF grant DMS-1216454 and ORNL. The work was partially performed at ORNL, which is managed by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qingdao University, Qingdao, 266071, Shandong, People’s Republic of China

    Gang Li

  2. Department of Mathematics, University of California Riverside, Riverside, CA, 92521, USA

    Yulong Xing

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  1. Gang Li
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Correspondence to Yulong Xing.

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Li, G., Xing, Y. Well-Balanced Discontinuous Galerkin Methods for the Euler Equations Under Gravitational Fields. J Sci Comput 67, 493–513 (2016). https://doi.org/10.1007/s10915-015-0093-5

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  • DOI: https://doi.org/10.1007/s10915-015-0093-5

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