Energy conserving and well-balanced discontinuous Galerkin methods for the Euler-Poisson equations in spherical symmetry

Abstract

This paper presents high-order Runge-Kutta (RK) discontinuous Galerkin methods for the Euler-Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to machine precision with carefully designed spatial and temporal discretizations. To achieve the well-balanced property, the numerical solutions are decomposed into equilibrium and fluctuation components which are treated differently in the source term approximation. One non-trivial challenge encountered in the procedure is the complexity of the equilibrium state, which is governed by the Lane-Emden equation. For total energy conservation, we present second- and third-order RK time discretization, where different source term approximations are introduced in each stage of the RK method to ensure the conservation of total energy. A carefully designed slope limiter for spherical symmetry is also introduced to eliminate oscillations near discontinuities while maintaining the well-balanced and total-energy-conserving properties. Extensive numerical examples -- including a toy model of stellar core-collapse with a phenomenological equation of state that results in core-bounce and shock formation -- are provided to demonstrate the desired properties of the proposed methods, including the well-balanced property, high-order accuracy, shock capturing capability, and total energy conservation.

References in corpus (8)

• Theory of Star Formation
• Three-dimensional core-collapse supernova simulated using a 15 $M_{\odot }$ progenitor
• Hydrodynamics of core-collapse supernovae and their progenitors
• Star Formation in Quasar Disk
• A New Way to Conserve Total Energy for Eulerian Hydrodynamic Simulations with Self-Gravity
• Three Dimensional Magnetohydrodynamical Simulations of Core Collapse Supernova
• An Extension of the Athena++ Framework for Fully Conservative Self-Gravitating Hydrodynamics
• thornado-hydro: a discontinuous Galerkin method for supernova hydrodynamics with nuclear equations of state

Cited by in corpus (2)

• Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks
• GenASiS: General Astrophysical Simulation System. II. Self-gravitating Baryonic Matter