A Metric Tensor Approach to Data Assimilation with Adaptive Moving Meshes

TL;DR

This paper introduces a metric tensor-based framework for adaptive moving meshes in data assimilation, improving the placement of mesh points near observations and enhancing the accuracy of ensemble-based DA methods.

Contribution

It develops a novel time-dependent mesh generation and localization scheme using metric tensors, tailored for ensemble data assimilation with adaptive meshes.

Findings

MT localization outperforms standard Gaspari-Cohn localization.

Adaptive meshes improve observation interpolation accuracy.

DG interpolation yields better results than linear methods in 2D cases.

Abstract

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.