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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P. Chandrashekar, C. Klingenberg, A second order well-balanced finite volume scheme for Euler equations with gravity. SIAM J. Sci. Comput. 37(3), B382–B402 (2015)
S. Chandrasekhar, An Introduction to the Study of Stellar Structure, vol. 2 (Courier Corporation, 1958)
P.V.F. Edelmann, Ph.D. thesis, Technische Universität München (2014)
M. Hosea, L. Shampine, Method of lines for time-dependent problems. Appl. Numer. Math. 20, 21 (1996)
R.J. LeVeque, D.S. Bale, Wave propagation methods for conservation laws with source terms, Hyperbolic Problems: Theory, Numerics, Applications (Birkhäuser, Basel, 1999), pp. 609–618
M.-S. Liou, J. Comput. Phys. 214, 137 (2006)
F. Miczek, Ph.D. thesis, Technische Universität München (2013)
P. Roe, J. Comput. Phys. 43, 357 (1981)
C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)
Y. Xing, C.-W. Shu, High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields. J. Sci. Comput. 54(2–3), 645–662 (2013)
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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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DOI: https://doi.org/10.1007/978-3-319-91545-6_12
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