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.

The computational model of a physical phenomenon is typically based on solving a particular partial
differential equation (PDE) with a numerical scheme. Numerical techniques to solve PDEs evolving in
time are most often based on a discretization of the underlying spatial domain. The resulting mesh
is generally fixed in time, but the needs of a given application may require the mesh itself to
change as the system evolves, adapting to the underlying physics

Two reasons that may lead to the use of an adaptive mesh are as follows: (1) for fluid problems, it is
sometimes preferable to pose the underlying PDEs in a Lagrangian, as opposed to Eulerian, frame or
(2) the model produces a specific structure, such as a front, shock wave or overflow, which is
localized in space. In case 1, the Lagrangian solver will naturally move the mesh with the
evolution of the PDE

Data assimilation (DA) is the process by which data from observations are assimilated into a
computational model of a physical system. There are numerous mathematical approaches, and associated
numerical techniques, for approaching this issue

Mesh adaptation brings significant challenges to DA. In particular, a time-varying mesh
may introduce difficulties in the gradient calculation within variational DA

Two specific pieces of work can be viewed as precursors of our methodology.

This work is further motivated by a specific application, namely performing ensemble-based DA for a
new class of computational models of sea ice

neXtSIM is solved on a 2-D unstructured triangular adaptive moving mesh based on a
Lagrangian solver that propagates the mesh of the model in time along with the motion of the sea ice

The specific DA methodology we develop for adaptive mesh problems is driven by the considerations of neXtSIM. The remeshing in neXtSIM, and the consequent change in the state vector’s dimension, is addressed in our assimilation scheme by the introduction of a reference mesh. The latter represents a common mesh for forming the error covariance matrix from the ensemble members. The question then arises as to whether this common mesh is used to propagate each individual ensemble member forward in time. From the viewpoint of neXtSIM, however, continuing with the reference mesh, common to all members, could throw away valuable physical information. In fact, the use of a Lagrangian solver in neXtSIM assures that the mesh configurations are naturally attuned to the physical evolution of the ice. For this reason, we make the critical methodological decision to map back to the meshes of the individual ensemble members after the assimilation step. The Lagrangian solver in the model is thus the primary determinant of the mesh configuration used in each forecast step. The reference mesh is only used in a temporary capacity to afford a consistent update at the assimilation step.

In this paper, we construct a 1-D setup designed to capture the core issues that neXtSIM
presents for the application of an ensemble-based DA scheme. We perform experiments using both
Eulerian (where the observation locations are fixed) 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 situation that
is very common in physical science, particularly in the DA applications to the geosciences

Our core strategy is to introduce a fixed reference mesh onto which the meshes of the individual
ensemble members are mapped. A key decision is how refined the fixed reference mesh be made. There
are two natural choices here: (a) one that has

There have been other recent studies aimed at tackling the issue of DA on adaptive and/or moving
meshes.

In summary,

The paper is organized as follows: in Sect.

This paper focuses on a physical model describing the evolution of a scalar quantity,

Solving Eq. (

A further key feature of neXtSIM as a computational model is that it incorporates a remeshing
procedure. As a result, it is different from the usual problems considered in the adaptive mesh
literature

We build here a 1-D periodic adaptive moving mesh that retains the key features of the neXtSIM's 2-D mesh in being Lagrangian and including remeshing.

For a fixed time, a mesh is given by a set of points

The mesh will evolve following the Lagrangian dynamics associated with the solution of the PDE
(Eq.

The neXtSIM model adopts an alternative strategy that bases the prediction of the mesh at time

In neXtSIM, the coupled system, which includes the mesh and the physical model, is solved in
three successive steps. (1) The mesh solver is integrated to obtain the mesh points at

In the first step, the movement of the mesh nodes is determined by the behavior of the physical
model, which is a special case of the mesh being adaptive. In particular, the dynamics of the
physical model can lead to the emergence of sharp fronts or other localized structures. These
features can then be better resolved through the finer grid that now covers the relevant region,
which is the usual motivation behind the use of adaptive meshes in general. This may result,
however, in the allocation of a significant quantity of the total number of nodes to a small portion
of the computational domain. Such a convergence of multiple nodes in a small area can lead to a
reduction of the computational accuracy in other areas of the model domain and to the increase in
the computational cost, as smaller time steps will be required. In the case of a mesh made up of
triangular elements, as in neXtSIM, those may get too distorted, leading again to a reduction of the
numerical accuracy of the finite element solution

Adaptive mesh methods often invoke a mesh density function in Eq. (

In the 1-D models described in Sect.

We now view the mesh points

When an invalid mesh is encountered as a result of the advection process, a new valid mesh is
created that preserves as many of these nodes as possible. A validity check is made at each
computational time step. The remeshing is accomplished by looping through the nodes

For each

The result of the remeshing will be a new mesh reordered according to Eq. (

An illustration of the remeshing process with

The remeshing algorithm, with

Figure

An illustration of adaptive moving mesh over time solving Burgers' equation (see
Sect.

Since both the physical value(s) representing the system and the mesh on which the PDE is solved are
evolved, we represent both in the state vector. The dimension of the state vector is then

The model will encompass all the algebraic relations of the computation, including the mesh
advancement and remeshing. It need not be defined for every

The model operates between observation times. If we set

We introduce a modification of the EnKF

The EnKF, originally introduced by

The challenge in implementing an EnKF on an adaptive moving mesh model with remeshing is that the
dimension of the state vector will be potentially different for each ensemble member. This is
addressed by

The location of the nodes and their total number are bound to change with time and across ensemble members: each member now provides a distinct discrete representation of the underlying continuous physical process based on a different number of differently located sample points. The individual ensemble members have to be intended now as samples from a different partition of the physical system's phase space, and they do not provide a statistically consistent sampling of the discrete-in-space uncertainty distribution. This is reflected in practice by the fact that the members cannot be stored column-wise any longer to form ensemble matrices, and thus the matrix computations involved in the EnKF analysis to evaluate the ensemble-based mean and covariance cannot be performed.

On the other hand, on the reference mesh, the members are all samples from the same discrete distribution and can thus be used to compute the ensemble-based mean and covariance. The entire EnKF analysis process is carried out on this fixed reference mesh, and the results are then mapped back to the individual ensemble meshes. This procedure amounts to the addition of two steps on top of those in the standard EnKF. First, we map each ensemble member from its adaptive moving mesh to an appropriate fixed uniform mesh and perform the analysis. Then, the updated ensemble members are mapped back to their adaptive moving meshes, providing the ensemble for the next forecast step.

The process is summarized schematically in Fig.

Illustration of the analysis cycle in the proposed EnKF method for adaptive moving mesh
models. In

We divide the physical domain

While we are, in principle, free to choose the fixed reference mesh arbitrarily, it makes sense to tailor it to the application under consideration. We choose to define the resolution of this fixed uniform mesh based on the maximum and minimum possible resolution of the individual adaptive moving meshes in the ensemble. The resolution range in the adaptive moving mesh reflects the computational constraints adapted to the specific physical problem: it therefore behooves us to bring these constraints into the definition of the fixed mesh for the analysis.

The high-resolution fixed reference mesh (HR) will be obtained by setting

Note that the hypothesis

The mapping will take a state vector

We denote the mapping as

To set the

For the EnKF, we will also need the map that omits the mesh points in the fixed reference mesh,

In the EnKF analysis, we will denote

The observations will be of physical values (

Thus, we can eventually define the state vector on

After mapping all the ensemble members onto the dedicated fixed reference mesh (either the high- or
the low-resolution one), the stochastic EnKF can be applied in the standard way. This is step

Model outputs are confronted with the observations at the end of every analysis interval and are
stored in the observation vector,

In the stochastic EnKF

When applied to large dimensional systems for which

The updated analysis ensemble in Eq. (

After the analysis, the update on the fixed reference mesh has to be mapped back onto the individual
adaptive moving meshes of the ensemble members. In the forward mapping step

Each analysis ensemble member

In summary, each ensemble member after the analysis step will have the form

The process steps

Schematic illustration of the DA cycle on the high-resolution

Let consider first the HR case of Fig.

Similarly, Fig.

We note a key aspect of our methodological choice: the ratio of the remeshing criteria

Our aim is to test the modified EnKF methodology described in Sect.

The first numerical model is the diffusive version of Burgers' equation

As second model, we use an implementation of the Kuramoto–Sivashinsky equation

Two “nature runs” are obtained, one for each model, by integrating them on a high-resolution fixed
uniform mesh. For both models, the meshes for the nature are intentionally chosen to be of at least
the same resolution of the HR fixed uniform reference mesh of the analysis. The size of the nature
run's mesh for the BGM is

We have limited the time length of the simulations in BGM to

With the given choice of the viscosity, KSM is not as dissipative as BGM and simulations can be run for
longer. KSM is initialized using

Numerical solutions of Burgers' and Kuramoto–Sivashinsky equations. The solutions are computed on an uniform fixed mesh and represent the nature run from which synthetic Eulerian and Lagrangian observations are sampled.

Synthetic Eulerian and Lagrangian observations are sampled from the nature run. Eulerian
observations are always collected at the same, fixed-in-time locations of the domain. We assume
that Eulerian observers are evenly distributed along the one-dimensional domain (i.e., observations
are at equally spaced locations) and that their total number is constant, so the number of
observations at time step

Observations sampled from the BGM nature run (see Fig.

In the experiments that follow, we chose to deploy as many Lagrangian observers at

Experimental setup parameters:

The experiments are compared by looking at the root-mean-square error (RMSE) of the ensemble mean
(with respect to the nature run) and the ensemble spread. Since the analysis is performed on either
the HR or the LR fixed mesh, the computation of the RMSE and spread is done on the mesh resulting
from their intersection. Given that we have chosen the remeshing criteria in both models such that

We present the results in three subsections. In Sect.

In this section, the experiments using BGM are presented. In order to calculate the base error due
to the choice of the specific fixed reference mesh, HR or LR, and the resulting mapping procedures,
we first perform an ensemble run without assimilation. This DA-free ensemble run is subject to all
of the steps described in Fig.

Figure

Time evolution of the forecast RMSE (solid line) and spread (

In the DA experiments, we study the sensitivity of the EnKF to the ensemble size, inflation factor and initial size of the adaptive moving meshes. Recall that the ensemble members are all given the same uniform mesh at the initial time; however these meshes will then inevitably evolve into a different, generally non-uniform mesh for each member. We remark that the three parameters under consideration are all interdependent, and a proper tuning would involve varying them simultaneously, which would make the number of experiments grow too much. To reduce the computational burden, we opted instead to vary only one at a time while keeping the other two fixed.

The results of this tuning are displayed in Fig.

Time mean of the RMSE of the analysis ensemble mean (solid line) and ensemble spread
(

In the case of the sensitivity to the ensemble size (Fig.

We therefore set

Finally, in Fig.

The results of the tuning experiments of Fig.

Ensemble size (

Time evolution of the RMSE (solid line) and spread (

This section shows the same type of results as in the previous section, this time being applied to the
KSM. We begin by evaluating the errors related to the mapping on the HR and LR case by running a
DA-free ensemble; results are shown in Fig.

Same as Fig.

As opposed to what is observed in Fig.

Figure

Same as Fig.

In Fig.

Same as Table

Figure

Up to this point, we have only utilized Eulerian observations. Using the optimal setup presented in
the previous sections, we now assess the impact of different observation types, i.e., Eulerian or
Lagrangian (see Figs.

Figure

At first sight, one can infer from Fig.

Same as Fig.

Time evolution of the RMSE until

We propose a novel methodology to perform ensemble data assimilation with computational models that use non-conservative adaptive moving mesh. Meshes of this sort are said to be adaptive because their node locations adjust to some prescribed rule that is intended to improve model accuracy. We have focused here on models with a Lagrangian solver in which the nodes move following the model's velocity field. They are said to be non-conservative because the total number of nodes in the mesh can itself change when the mesh is subject to remeshing. We have considered the case in which remeshing avoids having nodes too close or too far apart than given tolerance distances; in practice the tolerances define the set of valid meshes. When an invalid mesh appears through integration, it is then remeshed and a valid one is created.

The major challenge for ensemble data assimilation stands in that the dimension of the state space changes in time and differs across ensemble members, impeding the normal ensemble-based operations (i.e., matrix computations) at the analysis update. To overcome this issue, we have added in our methodology one forward and one backward mapping step before and after the analysis, respectively. This mapping takes all the ensemble members onto a fixed, uniform reference mesh. On this mesh, all ensemble members have the same dimension and are defined onto the same spatial mesh; thus the assimilation of data can be performed using standard EnKF approaches. We have used the stochastic EnKF, but the approach can be easily adapted to the use of a square-root EnKF. After the analysis, the backward mapping returns the updated values to the individual, generally different and non-uniform meshes of the respective ensemble members.

We consider two cases: a high-resolution and a low-resolution fixed uniform reference mesh. The
essential property is that their resolution is determined by the remeshing tolerances

We tested our modified EnKF using two 1-D models, the Burgers and Kuramoto–Sivashinsky equations. A set of sensitivity tests is carried through some key model and DA setup parameters: the ensemble size, inflation factor and initial mesh size. We considered two types of observations: Eulerian and Lagrangian. It is shown that, in general, a high-resolution fixed reference mesh improves the estimate more than a low-resolution fixed reference mesh. Whereas this might indeed be expected, our results also show that a low-resolution reference mesh affords a very high level of accuracy if the EnKF is properly tuned for the context. The use of a low-resolution fixed mesh has the obvious advantage of a lower computational burden, given that the size of the matrix operations is to be implemented at the analysis step scales with the size of the fixed reference mesh.

We then examined the impact of assimilating Lagrangian observations compared with Eulerian ones and have seen, in the context of Burgers equation, that the former improves the solution as much as the latter. The effectiveness of Lagrangian observers, despite being fewer in number than for the case of fixed, Eulerian observations, comes from their concentrating, where their information is most useful, i.e., within the sharp single (shock-like) front of the Burgers solution.

In this work, we have focused on the design of the strategy and, for the sake of clarity, have focused only on updating the physical quantities, while the locations of the ensemble mesh nodes were left unchanged. A natural extension of this study is to subject both the model physical variables and the mesh locations to the assimilation of data. Both would then be updated at the analysis time, and this is currently under investigation.

This paper is part of a longer-term research effort aimed at developing suitable EnKF strategies for a next-generation 2-D sea-ice model of the Arctic Ocean, neXtSIM, which solves the model equations on a triangular mesh using finite element methods and a Lagrangian solver. The velocity-based mesh movement and remeshing procedure that we have built into our 1-D model scenarios were formulated with the aim of mimicking those specific aspects of neXtSIM. In terms of our different types of observations, the impact of Eulerian and Lagrangian observational data was studied in light of observations gathered by satellites and drifting buoys, respectively, which are two common observing tools for Arctic sea ice.

Such a 2-D extension is, however, a non-trivial task for a number of fundamental reasons. First of all, given the triangular unstructured mesh in neXtSIM, we cannot straightforwardly define an ordering of the nodes on the adaptive moving mesh, as is done in the 1-D case considered here. As a consequence, the determination of a fixed reference mesh might not be linked to the remeshing criteria in the same straightforward way. However, it is still possible to define a high- or low-resolution fixed reference mesh with respect to the mesh of neXtSIM, since the remeshing in neXtSIM is mainly used to keep the initial resolution throughout the integration. Secondly, the models considered in this study are proxies of continuous fluid flows, whereas the rheology implemented in neXtSIM treats the sea ice as a solid brittle material which results in discontinuities when leads and ponds form due to fracturing and ice melting; the Gaussian assumptions implicit in the EnKF formulation then need to be reconsidered. Nevertheless, the methodology presented in this study, and the experiments herein, confronts some of the key technical issues of the 2-D case. The current results in 1-D are encouraging regarding the applicability of the proposed modification of the EnKF to adaptive moving mesh models in two dimensions, and the extension to two dimensions is the subject of the authors' current research.

No data sets were used in this article.

The methodology has been developed by all of the authors. AA and CG coded and ran the experiments. The paper has been edited by all authors.

The authors declare that they have no conflict of interest.

The authors are thankful to Laurent Bertino for his comments and suggestions. We are thankful also to the editor and three anonymous referees for their constructive reviews.

Ali Aydoğdu, Alberto Carrassi and Pierre Rampal have been funded by the project REDDA (no. 250711) of the Norwegian Research Council. Alberto Carrassi and Pierre Rampal have also benefitted from funding from the Office of Naval Research project DASIM II (award N00014-18-1-2493). Colin T. Guider and Chris K. R. T. Jones have been funded by the Office of Naval Research award N00014-18-1-2204.

This paper was edited by Wansuo Duan and reviewed by three anonymous referees.