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arXiv 1306.2214astro-ph.SRcs.NAmath.NA
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用于恒星脉动计算的稳定高阶有限差分方案

Stable higher order finite-difference schemes for stellar pulsation calculations

Institut d’Astrophysique et Géophysique de l’Université de Liège(列日大学天体物理与地球物理研究所);Observatoire de Paris, LESIA, CNRS UMR 8109(巴黎天文台)

D. R. Reese

AI总结:

本文提出一种稳定高阶有限差分方案,解决恒星脉动计算中的网格漂移不稳定性问题,提升计算精度并保持灵活性。

摘要:

背景:计算恒星脉动需要足够的精度以匹配观测质量。许多当前脉动代码使用二阶有限差分方案,结合Richardson外推法以达到特征函数的四阶精度。尽管这种方法简单且稳健,但存在诸多缺点,因此四阶方案更具吸引力。一种稳健且简单的有限差分方案,可轻松应用于一维或二维恒星脉动代码中,因此是必需的。难点:在设置高阶有限差分方案用于恒星脉动时,所谓的网格漂移不稳定性是一个挑战。当前解决此问题的方法包括引入人工粘性或应用交错网格方法。然而,这些方法并不适用于特征值问题,尤其是涉及非耗散系统的,因为它们会大幅改变算子的谱,引入额外的自由参数,或在应用边界条件时导致复杂性。方法:本文提出了一种新的方法,受交错网格策略启发,能够消除此不稳定性并绕过上述困难。此外,这种方法有助于超收敛,即通过提升有限差分的精度一阶。结果:该新方法已被证明是准确的、对底层网格具有灵活性,并能消除网格漂移。

英文摘要:

Context: Calculating stellar pulsations requires a sufficient accuracy to match the quality of the observations. Many current pulsation codes apply a second order finite-difference scheme, combined with Richardson extrapolation to reach fourth order accuracy on eigenfunctions. Although this is a simple and robust approach, a number of drawbacks exist thus making fourth order schemes desirable. A robust and simple finite-difference scheme, which can easily be implemented in either 1D or 2D stellar pulsation codes is therefore required. Aims: One of the difficulties in setting up higher order finite-difference schemes for stellar pulsations is the so-called mesh-drift instability. Current ways of dealing with this defect include introducing artificial viscosity or applying a staggered grids approach. However these remedies are not well-suited to eigenvalue problems, especially those involving non-dissipative systems, because they unduly change the spectrum of the operator, introduce supplementary free parameters, or lead to complications when applying boundary conditions. Methods: We propose here a new method, inspired from the staggered grids strategy, which removes this instability while bypassing the above difficulties. Furthermore, this approach lends itself to superconvergence, a process in which the accuracy of the finite differences is boosted by one order. Results: This new approach is shown to be accurate, flexible with respect to the underlying grid, and able to remove mesh-drift.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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