A well-balanced scheme for the simulation tool-kit A-MaZe: implementation, tests, and first applications to stellar structure

¹ Centre de Recherche Astrophysique de Lyon UMR5574, Univ. Lyon, Ens de Lyon, Univ. Lyon1, CNRS, 69364 Lyon Cedex 07, France
e-mail: doris.folini@ens-lyon.fr
² College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QL, UK
³ Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303, USA
⁴ Seminar of Applied Mathematics, ETH-Zürich, Switzerland

Received: 3 September 2018
Accepted: 30 August 2019

Abstract

Characterizing stellar convection in multiple dimensions is a topic at the forefront of stellar astrophysics. Numerical simulations are an essential tool for this task. We present an extension of the existing numerical tool-kit A-MaZe that enables such simulations of stratified flows in a gravitational field. The finite-volume based, cell-centered, and time-explicit hydrodynamics solver of A-MaZe was extended such that the scheme is now well-balanced in both momentum and energy. The algorithm maintains an initially static balance between gravity and pressure to machine precision. Quasi-stationary convection in slab-geometry preserves gas energy (internal plus kinetic) on average, despite strong local up- and down-drafts. By contrast, a more standard numerical scheme is demonstrated to result in substantial gains of energy within a short time on purely numerical grounds. The test is further used to point out the role of dimensionality, viscosity, and Rayleigh number for compressible convection. Applications to a young sun in 2D and 3D, covering a part of the inner radiative zone, as well as the outer convective zone, demonstrate that the scheme meets its initial design goal. Comparison with results obtained for a physically identical setup with a time-implicit code show qualitative agreement.