Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry
Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry
复制标题
球对称中欧拉泊松方程的能量守恒且平衡良好的间断伽辽金方法
DOI：
10.1093/mnras/stac1257
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发表时间：
2022
影响因子：
4.8
通讯作者：
Endeve, Eirik
中科院分区：
文献类型：
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作者：
Zhang, Weijie;Xing, Yulong;Endeve, Eirik
提出了求解球对称Euler-Poisson方程的高阶Runge-Kutta（RK）间断Galerkin方法。该方案可以保持一个一般多变的平衡状态，并实现总能量守恒的机器精度精心设计的空间和时间离散。为了实现良好的平衡属性，数值解被分解为平衡和波动分量，在源项近似中被不同地对待。在该过程中遇到的一个重要挑战是平衡态的复杂性，这是由莱恩-埃姆登方程。对于总能量守恒，我们提出了二阶和三阶RK时间离散化，其中在RK方法的每个阶段引入不同的源项近似，以确保总能量守恒。一个精心设计的球对称的斜率限制器也被引入，以消除附近的不连续性的振荡，同时保持良好的平衡和总能量守恒的属性。广泛的数值例子-包括玩具模型的恒星核心崩溃的现象学状态方程，导致在核心反弹和冲击形成-提供演示所需的性能的方法，包括良好的平衡性能，高阶精度，冲击捕获能力，和总能量守恒。
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 that 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.