A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime
A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime
复制标题
史瓦西时空中可压缩流体的保几何有限体积法
DOI：
10.4208/cicp.291212.160913a
复制
发表时间：
2014
影响因子：
3.7
通讯作者：
Hasan Makhlof
中科院分区：
文献类型：
--
作者：
P. Floch;Hasan Makhlof
本文考虑了Schwarzschild时空外连通域上的球对称理想流体流动的相对论性Euler方程，并引入了一种由几何公式推导出的有限体积方法（并且因此在离散化级别考虑几何形状）并且是良好平衡的，在这个意义上，它保留了稳定的解决方案，欧拉方程的弯曲几何正在考虑。为了制定我们的方法，我们首先推导出一个封闭的公式，描述所有的稳定和球对称的解决方案的欧拉方程提出的史瓦西时空。其次，我们描述了一个几何保持，有限体积法，这是基于从家庭的稳定解决方案的欧拉系统。我们的计划是二阶精度，并根据需要，保持家庭的稳定的解决方案在离散水平。数值实验表明，所提出的方法的效率和鲁棒性，即使包含冲击波和非线性相互作用波图案的解决方案。作为应用，我们研究了扰动稳定解的后期渐近性，并证明了其收敛到另一个稳定解的后期时间，考虑到扰动的整体效果。
We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild space-time, and we introduce a version of the finite volume method which is formulated from the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions to the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based from the family of steady solutions to the Euler system. Our scheme is second-order accurate and, as required, preserves the family of steady solutions at the discrete level. Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave patterns. As an application, we investigate the late-time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution, taking the overall effect of the perturbation into account.