Abstract. Numerical models solved on adaptive moving meshes have become increasingly prevalent in recent years. Motivating problems include the study of fluids in a Lagrangian frame and the presence of highly localized structures such as shock waves or interfaces. In the former case, Lagrangian solvers move the nodes of the mesh with the dynamical flow; in the latter, mesh resolution is increased in the proximity of the localized structure. Mesh adaptation can include remeshing, a procedure that adds or removes mesh nodes according to specific rules reflecting constraints in the numerical solver. In this case, the number of mesh nodes will change during the integration and, as a result, the dimension of the model's state vector will not be conserved. This work presents a novel approach to the formulation of ensemble data assimilation (DA) for models with this underlying computational structure. The challenge lies in the fact that remeshing entails a different state space dimension across members of the ensemble, thus impeding the usual computation of consistent ensemble-based statistics. Our methodology adds one forward and one backward mapping step before and after the ensemble Kalman filter (EnKF) analysis, respectively. This mapping takes all the ensemble members onto a fixed, uniform reference mesh where the EnKF analysis can be performed. We consider a high-resolution (HR) and a low-resolution (LR) fixed uniform reference mesh, whose resolutions are determined by the remeshing tolerances. This way the reference meshes embed the model numerical constraints and are also upper and lower uniform meshes bounding the resolutions of the individual ensemble meshes. Numerical experiments are carried out using 1-D prototypical models: Burgers and Kuramoto–Sivashinsky equations and both Eulerian and Lagrangian synthetic observations. While the HR strategy generally outperforms that of LR, their skill difference can be reduced substantially by an optimal tuning of the data assimilation parameters. The LR case is appealing in high dimensions because of its lower computational burden. Lagrangian observations are shown to be very effective in that fewer of them are able to keep the analysis error at a level comparable to the more numerous observers for the Eulerian case. This study is motivated by the development of suitable EnKF strategies for 2-D models of the sea ice that are numerically solved on a Lagrangian mesh with remeshing.

中文翻译：

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使用自适应、非保守、移动网格模型进行数据同化

摘要。近年来，在自适应移动网格上求解的数值模型变得越来越普遍。激励问题包括研究拉格朗日框架中的流体以及高度局部化结构的存在，例如冲击波或界面。在前一种情况下，拉格朗日求解器随着动力流移动网格的节点；在后者中，局部结构附近的网格分辨率增加。网格自适应可以包括重新网格划分，这是一种根据反映数值求解器中约束的特定规则添加或删除网格节点的过程。在这种情况下，网格节点的数量会在积分过程中发生变化，因此模型状态向量的维数将不守恒。这项工作提出了一种新的方法来为具有这种基础计算结构的模型制定集成数据同化 (DA)。挑战在于重新网格划分需要跨集成成员的不同状态空间维度，从而阻碍一致的基于集成的统计数据的通常计算。我们的方法分别在集成卡尔曼滤波器 (EnKF) 分析之前和之后添加了一个向前和一个向后映射步骤。这种映射将所有的整体成员带到一个固定的、统一的参考网格上，在那里可以执行 EnKF 分析。我们考虑高分辨率 (HR) 和低分辨率 (LR) 固定均匀参考网格，其分辨率由重新网格化公差决定。通过这种方式，参考网格嵌入了模型数值约束，并且也是限制单个整体网格分辨率的上下均匀网格。数值实验使用一维原型模型进行：Burgers 和 Kuramoto-Sivashinsky 方程以及欧拉和拉格朗日的综合观测。虽然 HR 策略通常优于 LR，但可以通过优化数据同化参数来显着减少它们的技能差异。LR 案例在高维度上很有吸引力，因为它的计算负担较低。拉格朗日观测被证明是非常有效的，因为在欧拉情况下，能够将分析误差保持在与更多观测者相当的水平的观测者较少。