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Well-balanced finite volume schemes for nearly steady adiabatic flows
Journal of Computational Physics ( IF 4.1 ) Pub Date : 2020-08-31 , DOI: 10.1016/j.jcp.2020.109805
L. Grosheintz-Laval , R. Käppeli

We present well-balanced finite volume schemes designed to approximate the Euler equations with gravitation. They are based on a novel local steady state reconstruction. The schemes preserve a discrete equivalent of steady adiabatic flow, which includes non-hydrostatic equilibria. The proposed method works in Cartesian, cylindrical and spherical coordinates. The scheme is not tied to any specific numerical flux and can be combined with any consistent numerical flux for the Euler equations, which provides great flexibility and simplifies the integration into any standard finite volume algorithm. Furthermore, the schemes can cope with general convex equations of state, which is particularly important in astrophysical applications. Both first- and second-order accurate versions of the schemes and their extension to several space dimensions are presented. The superior performance of the well-balanced schemes compared to standard schemes is demonstrated in a variety of numerical experiments. The chosen numerical experiments include simple one-dimensional problems in both Cartesian and spherical geometry, as well as two-dimensional simulations of stellar accretion in cylindrical geometry with a complex multi-physics equation of state.



中文翻译:

几乎稳定的绝热流的均衡有限体积方案

我们提出了平衡良好的有限体积方案,旨在用引力近似欧拉方程。它们基于新颖的局部稳态重建。该方案保留了稳态绝热流的离散等效项,其中包括非静水平衡。所提出的方法适用于笛卡尔,圆柱和球面坐标。该方案不依赖于任何特定的数值通量,可以与用于Euler方程的任何一致的数值通量组合,从而提供了极大的灵活性,并简化了对任何标准有限体积算法的集成。此外,该方案可以应对状态的一般凸方程,这在天体物理应用中尤其重要。给出了该方案的一阶和二阶精确版本以及它们对几个空间维度的扩展。在各种数值实验中证明了均衡方案比标准方案优越的性能。选择的数值实验包括笛卡尔和球形几何中的简单一维问题,以及具有复杂的多物理场状态方程的圆柱几何中恒星积聚的二维模拟。

更新日期:2020-08-31
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