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.
翻译:本文介绍了在球度对称中,Euler-Poisson方程式的高顺序龙格-库塔(RK)不连续的Galerkin(RK)方法。这个方法可以保持一般的多元平衡状态,并实现总节能,直至机精度,同时仔细设计空间和时间分解。为了实现平衡属性,数字解决方案被分解成平衡和波动部分,在源术语近距离中处理不同。程序中遇到的一个非三进制的挑战就是平衡状态的复杂性,由道-埃姆登方程式管理。在总体节能方面,我们提出第二和第三级的RK时间分解,在RK方法的每个阶段都采用不同的源词近似值,以确保总节能。还引入了精心设计的球系对称斜坡度限制,以消除接近不连续状态的振荡,同时保持平衡和全节能特性。广泛的数字实例 -- 包括一个具有全成型的恒心核心核心-核心-核心-稳定状态的模型,包括拟议的休克-休克-测结果的形成,包括拟议的高震力-稳定的休测-提供高震力的状态的核心-测测测。