Adaptive moving spatial meshes are useful for solving physical models given by time-dependent partial differentialequations. However, special consideration must be given when combining adaptive meshing procedures with ensemble-based data assimilation (DA) techniques. In particular, we focus on the case where each ensemble member evolvesindependently upon its own mesh and is interpolated to a common mesh for the DA update. This paper outlines aframework to develop time-dependent reference meshes using locations of observations and the metric tensors (MTs)or monitor functions that define the spatial meshes of the ensemble members. We develop a time-dependent spatiallocalization scheme based on the metric tensor (MT localization). We also explore how adaptive moving mesh tech-niques can control and inform the placement of mesh points to concentrate near the location of observations, reducingthe error of observation interpolation. This is especially beneficial when we have observations in locations that wouldotherwise have a sparse spatial discretization. We illustrate the utility of our results using discontinuous Galerkin(DG) approximations of 1D and 2D inviscid Burgers equations. The numerical results show that the MT localizationscheme compares favorably with standard Gaspari-Cohn localization techniques. In problems where the observationsare sparse, the choice of common mesh has a direct impact on DA performance. The numerical results also demonstratethe advantage of DG-based interpolation over linear interpolation for the 2D inviscid Burgers equation.

中文翻译：

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自适应移动网格数据同化的度量张量方法

自适应移动空间网格可用于求解由瞬态偏微分方程给出的物理模型。但是，在将自适应网格划分程序与基于集成的数据同化 (DA) 技术相结合时，必须给予特殊考虑。特别是，我们关注每个集成成员独立于其自己的网格演化并被插入到公共网格以进行 DA 更新的情况。本文概述了使用观测位置和度量张量 (MT) 或定义集合成员空间网格的监控函数来开发时间相关参考网格的框架。我们开发了一种基于度量张量（MT 定位）的时间相关空间定位方案。我们还探讨了自适应移动网格技术如何控制和通知网格点的放置，以集中在观测位置附近，减少观测插值的误差。当我们在具有稀疏空间离散化的位置进行观察时，这尤其有益。我们使用一维和二维无粘伯格斯方程的不连续 Galerkin(DG) 近似来说明我们的结果的效用。数值结果表明，MT 定位方案优于标准的 Gaspari-Cohn 定位技术。在观察稀疏的问题中，公共网格的选择对 DA 性能有直接影响。数值结果还证明了基于 DG 的插值优于线性插值的二维无粘性 Burgers 方程。