Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers

Abstract

Accurate simulations of flows in stellar interiors are crucial to improving our understanding of stellar structure and evolution. Because the typically slow flows are merely tiny perturbations on top of a close balance between gravity and the pressure gradient, such simulations place heavy demands on numerical hydrodynamics schemes. We demonstrate how discretization errors on grids of reasonable size can lead to spurious flows orders of magnitude faster than the physical flow. Well-balanced numerical schemes can deal with this problem. Three such schemes were applied in the implicit, finite-volume Seven-League Hydro (SLH) code in combination with a low-Mach-number numerical flux function. We compare how the schemes perform in four numerical experiments addressing some of the challenges imposed by typical problems in stellar hydrodynamics. We find that the $\alpha$-$\beta$ and deviation well-balancing methods can accurately maintain hydrostatic solutions provided that gravitational potential energy is included in the total energy balance. They accurately conserve minuscule entropy fluctuations advected in an isentropic stratification, which enables the methods to reproduce the expected scaling of convective flow speed with the heating rate. The deviation method also substantially increases accuracy of maintaining stationary orbital motions in a Keplerian disk on long timescales. The Cargo-LeRoux method fares substantially worse in our tests, although its simplicity may still offer some merits in certain situations. Overall, we find the well-balanced treatment of gravity in combination with low Mach number flux functions essential to reproducing correct physical solutions to challenging stellar slow-flow problems on affordable collocated grids.

Accepted for publication in A&A

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• A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume/finite element scheme for the incompressible MHD equations
• Multidimensional Simulations of Core Convection
• Turbulent dynamo action and its effects on the mixing at the convective boundary of an idealized oxygen-burning shell
• Toward fully compressible numerical simulations of stellar magneto-convection with the RAMSES code
• Performance of high-order Godunov-type methods in simulations of astrophysical low Mach number flows
• Spectral Difference method with a posteriori limiting: II- Application to low Mach number flows
• DISPATCH methods: an approximate, entropy-based Riemann solver for ideal magnetohydrodynamics