Data assimilation using adaptive, non-conservative, moving mesh models

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 5 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 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 10 backward mapping step before and after the 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-(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 also are upper and lower uniform meshes bounding the resolutions of the individual ensemble meshes. Numerical experiments are carried out using 1D prototypical models: Burgers and Kuramoto-15 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 2D models of the sea-ice that are 20 numerically solved on a Lagrangian mesh with remeshing.and Lagrangian (where observation locations move with the flow) observations. We test the strategy for two well-known PDEs:the viscous Burgers and Kuramoto-Sivashinsky equation, whose associated computational models we refer to as BGM and KSM, respectively. The Burgers' equation, which can be viewed as modeling a one-dimensional fluid, is a canonical example for which a localized structure, in this case a shock wave, develops and an adaptive moving mesh will get denser near the shock front. The Kuramoto-Sivashinsky equation exhibits chaotic behavior and this provides a natural test-bed for DA in a dynamical Nonlin. Processes Geophys. ...

Ensemble Kalman filter for nonconservative moving mesh solvers with a joint physics and mesh location update

“…Dimension matching is done using the properties of the adaptive mesh scheme itself, via a partition of the domain with intervals of the same size as the proximity tolerance 1 , thus guaranteeing that each interval will contain at most one node. In the HR scheme developed in Aydogdu et al (2019), component pairing is done by shifting the nodes in each interval to their nearest interval boundaries and then interpolating new points to any empty interval boundaries. In this way, the HR method compares ensemble members on the same mesh, updating only the physical values of the nodes.…”

“…In this work, we are interested in adaptive meshes that evolve with the flow of a physical system and are nonconservative. We will make use of the same 1-d adaptive mesh scheme developed in Aydogdu et al (2019) as a prototype of the 2-or 3-d nonconservative adaptive meshes used in some modern numerical models, including the Lagrangian neXt generation Sea Ice Model, neXtSIM (Rampal et al, 2016;Rabatel et al, 2018;Cheng et al, 2020). The mesh itself is a 1-d mesh defined on the domain D = [0, L) with nodes {z 1 , z 2 , … , z N } ∈ D. It is assumed that 0 ≤ z i < z i + 1 < L and that the positions of the nodes satisfy criteria that define a valid mesh through two tolerance parameters 1 , 2 .…”

“…However, the node positions themselves are driven by the physical flow and as such can be considered time-dependent state variables. In this work, we consider updating the node locations by making use of the HR partitioning of the interval domain for the same models considered in Aydogdu et al (2019). The key difference between previous and current work is that we now augment our state vector with the node locations and update them in the analysis step.…”

“…Once the dimensions of the state vectors of each ensemble member have been matched, the EnKF can be applied in the usual way. As in Aydogdu et al (2019), we will use the stochastic version of the EnKF (Evensen, 2009). This choice does not influence how we design our schemes for AMMs.…”

“…In Aydoğdu et al . (2019), a fixed reference mesh is used in two 1‐d models for which the mesh evolves with the flow and undergoes a “remeshing” step, which injects new nodes should two be too far apart or removes nodes should two be too close together. This remeshing means that each ensemble member will likely have different numbers of points in different locations.…”