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A General Well-Balanced Finite Volume Scheme for Euler Equations with Gravity

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Theory, Numerics and Applications of Hyperbolic Problems I (HYP 2016)

Abstract

We present a second-order well-balanced Godunov-type finite volume scheme for compressible Euler equations with a gravitational source term. The scheme is designed to work for any hydrostatic equilibrium, which must be known á priori. It can be combined with any numerical flux function, time-stepping method, and grid topology. The scheme is based on the reconstruction of a special set of variables and a special source term discretization. We show the well-balanced property numerically for isothermal and polytropic equilibria in one and two dimensions using the Roe flux function and an explicit three-stage Runge–Kutta scheme. We demonstrate the superior resolution of small pressure perturbations of hydrostatic equilibria, down to an order \(10^{-10}\) and below compared to the hydrostatic background.

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Correspondence to Jonas P. Berberich .

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Berberich, J.P., Chandrashekar, P., Klingenberg, C. (2018). A General Well-Balanced Finite Volume Scheme for Euler Equations with Gravity. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 236. Springer, Cham. https://doi.org/10.1007/978-3-319-91545-6_12

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