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Well-Balanced Nodal Discontinuous Galerkin Method for Euler Equations with Gravity

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Abstract

We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss–Lobatto–Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.

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Acknowledgements

Praveen Chandrashekar thanks the Airbus Foundation Chair for Mathematics of Complex Systems at TIFR-CAM, Bangalore, for support in carrying out this work. Markus Zenk thanks the DAAD Passage to India program which supported his visit to Bangalore during which time part of this work was conducted.

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Authors and Affiliations

  1. TIFR Center for Applicable Mathematics, Bangalore, India

    Praveen Chandrashekar

  2. Department of Mathematics, University of Würzburg, Würzburg, Germany

    Markus Zenk

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  1. Praveen Chandrashekar
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  2. Markus Zenk
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Correspondence to Praveen Chandrashekar.

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Chandrashekar, P., Zenk, M. Well-Balanced Nodal Discontinuous Galerkin Method for Euler Equations with Gravity. J Sci Comput 71, 1062–1093 (2017). https://doi.org/10.1007/s10915-016-0339-x

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  • DOI: https://doi.org/10.1007/s10915-016-0339-x

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