A Metric Tensor Approach to Data Assimilation with Adaptive Moving Meshes

Adaptive moving spatial meshes are useful for solving physical models given by time-dependent partial differential equations. However, special consideration must be given when combining adaptive meshing procedures with ensemblebased data assimilation (DA) techniques. In particular, we focus on the case where each ensemble member evolves independently upon its own mesh and is interpolated to a common mesh for the DA update. This paper outlines a framework 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 spatial localization scheme based on the metric tensor (MT localization). We also explore how adaptive moving mesh techniques can control and inform the placement of mesh points to concentrate near the location of observations, reducing the error of observation interpolation. This is especially beneficial when we have observations in locations that would otherwise 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 localization scheme compares favorably with standard Gaspari-Cohn localization techniques. In problems where the observations are sparse, the choice of common mesh has a direct impact on DA performance. The numerical results also demonstrate the advantage of DG-based interpolation over linear interpolation for the 2D inviscid Burgers equation.

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