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Analysis of Schwarz waveform relaxation for the coupled Ekman boundary layer problem with continuously variable coefficients
Numerical Algorithms ( IF 1.7 ) Pub Date : 2021-07-07 , DOI: 10.1007/s11075-021-01149-y
Sophie Thery 1 , Florian Lemarié 1 , Eric Blayo 1 , Charles Pelletier 2
Affiliation  

In this paper, we present a global-in-time non-overlapping Schwarz method applied to the Ekman boundary layer problem. Such a coupled problem is representative of large-scale atmospheric and oceanic flows in the vicinity of the air-sea interface. Schwarz waveform relaxation (SWR) algorithms provide attractive methods for ensuring a “tight coupling” between the ocean and the atmosphere. However, the convergence study of such algorithms in this context raises a number of challenges. Numerous convergence studies of Schwarz methods have been carried out in idealized settings, but the underlying assumptions to make these studies tractable may prohibit them to be directly extended to the complexity of climate models. We illustrate this aspect on the coupled Ekman problem, which includes several essential features inherent to climate modeling while being simple enough for analytical results to be derived. We investigate its well-posedness and derive an appropriate SWR algorithm. Sufficient conditions for ensuring its convergence for different viscosity profiles are then established. Finally, we illustrate the relevance of our theoretical analysis with numerical results and suggest ways to improve the computational cost of the coupling. Our study emphasizes the fact that the convergence properties can be highly sensitive to some model characteristics such as the geometry of the problem and the use of continuously variable viscosity coefficients.



中文翻译:

具有连续可变系数的耦合 Ekman 边界层问题的 Schwarz 波形弛豫分析

在本文中,我们提出了一种应用于 Ekman 边界层问题的全局时间非重叠 Schwarz 方法。这种耦合问题代表了海气界面附近的大规模大气和海洋流动。Schwarz 波形弛豫 (SWR) 算法为确保海洋和大气之间的“紧密耦合”提供了有吸引力的方法。然而,在这种情况下对此类算法的收敛性研究提出了许多挑战。许多 Schwarz 方法的收敛研究是在理想化环境中进行的,但使这些研究易于处理的基本假设可能会阻止它们直接扩展到气候模型的复杂性。我们在耦合 Ekman 问题上说明了这一方面,它包括气候建模固有的几个基本特征,同时又足够简单,可以得出分析结果。我们研究了它的适定性并推导出一个合适的 SWR 算法。然后建立确保其收敛于不同粘度分布的足够条件。最后,我们说明了我们的理论分析与数值结果的相关性,并提出了提高耦合计算成本的方法。我们的研究强调了这样一个事实,即收敛特性可能对某些模型特征高度敏感,例如问题的几何形状和连续可变粘度系数的使用。然后建立确保其收敛于不同粘度分布的足够条件。最后,我们说明了我们的理论分析与数值结果的相关性,并提出了提高耦合计算成本的方法。我们的研究强调了这样一个事实,即收敛特性可能对某些模型特征高度敏感,例如问题的几何形状和连续可变粘度系数的使用。然后建立确保其收敛于不同粘度分布的足够条件。最后,我们说明了我们的理论分析与数值结果的相关性,并提出了提高耦合计算成本的方法。我们的研究强调了这样一个事实,即收敛特性可能对某些模型特征高度敏感,例如问题的几何形状和连续可变粘度系数的使用。

更新日期:2021-07-07
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