Strongly coupled nonlinear phenomena such as those described by Earth system models (ESMs) are composed of multiple component models with independent mesh topologies and scalable numerical solvers.
A common operation in ESMs is to remap or interpolate component solution fields defined on their computational mesh to another mesh with a different combinatorial structure and decomposition, e.g., from the atmosphere to the ocean, during the temporal integration of the coupled system. Several remapping schemes are currently in use or available for ESMs.
However, a unified approach to compare the properties of these different schemes has not been attempted previously. We present a rigorous methodology for the evaluation and intercomparison of remapping methods through an independently implemented suite of metrics that measure the ability of a method to adhere to constraints such as grid independence, monotonicity, global conservation, and local extrema or feature preservation. A comprehensive set of numerical evaluations is conducted based on a progression of scalar fields from idealized and smooth to more general climate data with strong discontinuities and strict bounds. We examine four remapping algorithms with distinct design approaches, namely ESMF Regrid

The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”) and Sandia National Laboratories. Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357, and Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.

The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative
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publicly, by or on behalf of the Government. The U.S. Department of Energy (DOE)
will provide public access to these results of federally sponsored
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Coupled multimodel simulations often involve high degrees of computationally complex workflows, and achieving consistently accurate solutions is strongly dependent on the choice of spatiotemporal numerical algorithms used to resolve the interacting scales in physical models. Rigorous spatial coupling between components in such systems involves field transformations and communication of data across multiresolution grids while preserving key attributes of interest such as global integrals and local features, which is usually referred to as the process of

Remapping algorithms, in general, compute the spatial interpolation or quasi-interpolation of field data that are defined on a source mesh (

Remapping packages developed for ESMs, such as the Community
Earth System Model (CESM)

Many of the production-ready remapping software implementations used in global climate simulations are typically based on first- and second-order conservative mesh-based schemes, with additional support for second-order nonconservative bilinear patch reconstructions

Remapping also occurs in other parts of a climate model, such as within the geophysical fluid dynamics solver of a component. Remapping strategies for tracer transport advection schemes such as the Semi-Lagrangian Inherently Conserving and Efficient (SLICE) scheme

While traditional mesh-based schemes requiring computation of overlay–exchange grids have the advantage of being inherently conservative

More recently, new mesh-based

As the number of available remapping algorithms grows, it becomes imperative to compare them under a unified framework to understand the properties of the schemes before applying them to real-world simulations. Additionally, while the computational cost is critically important for production runs, our intercomparison study specifically compares only the numerical performance of each algorithm under varying mesh topologies and field regularity, which are closely representative of those from a climate model such as E3SM. The presented protocol hence provides a systematic way to test and compare all existing and new remapping algorithms being developed for Earth system modeling.

The current study aims to better understand and document the key properties of the remapping schemes through the use of mesh, field, and scale resolution-independent metrics definitions. This intercomparison paper is organized as follows.

A detailed literature survey of the current state-of-the-art remapping schemes used in climate modeling and various related coupled physics problems, along with the relevant numerical background for four specific high-order remapping algorithms and their implementations, is presented in Sect.

Definitions for remapping metrics that evaluate field accuracy, global conservation, strict global bounds control, and feature dispersion are presented in Sect.

A sample workflow for computing and comparing various numerical metrics in a unified and unbiased fashion is featured in Sect.

Consolidated results from the intercomparison study applied to four competing remap algorithms on representative problems are shown in Sect.

Potential future research directions to extend the analysis presented in this work are described in Sect.

In general, for coupled simulations, we need to transfer a field

Finally, the accuracy or order of the remap typically depends on how the field

One of the simplest data transfer methods is to use piecewise interpolation functionals.
This approach is particularly convenient if

More generally, the remap methods designed for scattered data and cell-to-cell transfers are closely
related to the

More recently,

Typically, a remap method can and should be independent of the discretization
methods used in physics models. In this context, the method may be nonconservative or conservative with respect to certain properties of the reconstruction.
A major disadvantage of the

A common approach to overcome
this issue is to enforce

Concurrently, when applying high-order remapping methods, discontinuities in the function defined on

The deficiency in linear mapping approaches arises from the fact that they are only dependent on

In this vein, the techniques for resolving Gibbs phenomena can be classified as

Alternatively, rather than imposing a weakly nonlinear post-processing filter, using fully nonlinear remap schemes can be an option when the computational cost of the solution transfer is not the dominant factor in the simulation. Such nonlinear remap schemes typically use optimization-based remap (OBR) procedures

Additionally, mimetic schemes that use compatible function spaces

A common theme across all of the remapping methods described in this paper is that they utilize some variants of the least squares approximations (also called quasi-interpolation) in their computational kernels.
To illustrate the idea, let us consider a function

Among the standard conservative remapping and high-order reconstruction strategies introduced in Sect.

We consider two specific implementations that provide conservative remapping capability for production ESMs.

ESMF: the Earth System Modeling Framework's

TempestRemap: conservative, consistent, and monotone remapper with higher-order

Both ESMF Regrid and TempestRemap provide the remapping capability for scalar fields defined on

Note that the SCRIP library

Reconstruction methods that do not directly utilize the topological information about the underlying mesh layout are

Future studies could also include the comparison of MMLS

The final category includes the mesh-based remappers that do not require computation of an intersection mesh between

Other opportunities for comparison in this category include using reconstructed climate data with the conservative bilinear algorithm and patch reconstructions in ESMF or bicubic interpolations available in Yet Another Coupler (YAC)

These algorithms and their implementations span a range of remapping techniques, including those currently used in production runs and those that can potentially be used in the future given the availability of open-source software. Further details regarding the numerics and the software tools for each of the schemes are provided in the following subsections.

The Earth System Modeling Framework (ESMF,

In the current intercomparison study, we utilized the command-line applications installed with ESMF (version 8.1) to generate and apply interpolation weights from the command line using NetCDF files. Using this

TempestRemap (

In TempestRemap, the procedure used to generate remapping weights for FV discretizations consists of two primary operations

A common approach for the integration operator has been to construct a potential function with divergence equal to the scalar field being integrated and then use the divergence theorem to transform integrals over mesh faces into line integrals around the face

The generalized moving least squares (GMLS) method extends the moving least squares (MLS) technique from approximation of point values to approximation of arbitrary linear functionals

High-order accurate approximations cannot be achieved with traditional MLS schemes using function spaces described by Raviart–Thomas (

Traditionally, GMLS uses a basis that is defined as a function of the spatial dimension from which a point cloud is sampled. However, in this work, reconstruction of functions sampled on a manifold permits generating a compact stencil in a local coordinate chart, which is one dimension smaller

As is true for many other regression schemes, GMLS is not inherently conservative to machine precision, but rather it is “conservative” to discretization precision. In other words, the degree to which it violates conservation is discretization-dependent and generally vanishes with refinement. However, such weak conservation notions may not be deemed satisfactory for climate modeling applications for which exact global conservation has a history of being demanded and valued. In such cases, GMLS remap should be followed by a post-process filter to restore global conservation to machine precision. In this paper, we use either the GMLS or GMLS-CAAS notations to indicate whether the CAAS routine has been used as a post-processing filter after each remap step in order to restore global conservation or global bounds, along with an attempt to improve local property preservation.

Similar to the overlay-mesh-based, ESMF, and TempestRemap offline remappers described in Sect.

There are several key motivations for using GMLS to perform field remapping for ESMs. These include mesh topology independence, as well as flexibility in the choice of the sampling functional and the target operator, thereby enabling remap of nontraditional degrees of freedom that may be defined on the vertices, edges, or faces of

Unlike the preceding remapping techniques, WLS-ENOR is a non-overlay mesh-based technique in that it uses the mesh for computing numerical integration, but it does not require an overlay mesh (although it has the option to use the overlay mesh if available). More specifically, WLS-ENOR utilizes adaptive quadrature rules with

The second component in WLS-ENOR is the detection and resolution of discontinuities. In particular, WLS-ENOR can detect

The third component in WLS-ENOR is an adaptive quadrature technique, which is enabled when the target mesh is significantly coarser than the source mesh. In this case, simply sampling the function at the quadrature points of a target cell may miss some important local features on the source mesh, especially near discontinuities. To overcome this loss of information, one could use the overlay mesh as in TempestRemap. This approach may be ideal in terms of accuracy and conservation, but it introduces complications when the elements have curved edges. Although WLS-ENOR implementation supports this option, in this comparative work, we use a non-overlay-based version of WLS-ENOR that utilizes

The current implementation of the WLS-ENOR algorithm uses MATLAB, with which the core components were converted into C using “MATLAB Coder” (version 4.2). An open-source C++ implementation is currently underway and will be released in the future for both node-to-node and cell-to-cell field transfers.

Solution remapping on unstructured meshes is a complex process, and it is critical to satisfy several key properties to ensure that the transferred field data between components do not introduce unbounded and nonphysical error modes. In order to compare different remapping schemes in an unbiased framework, we introduce five primary categories under which the comparison metrics can be grouped.

Given a continuous field

A crucial factor for the success and broader usability of a general remapping algorithm in ESMs is the ability to produce mesh-independent numerical behavior that is robust for any pair of structured or unstructured meshes. In other words, remapping algorithms need to be general and without approximations targeted at specific topological elements.

In the current work, we utilized three different meshes of varying resolutions. Specifically, we present analysis performed to compare remapping schemes using the cubed-sphere (CS), quasi-uniform Voronoi (MPAS), regular latitude–longitude (RLL) meshes on both quasi-uniform and regionally refined meshes (RRMs).
Some sample meshes used in the study are shown in Fig.

A depiction of the five meshes studied in validation, unit testing, and intercomparison of the regridding schemes.

Accuracy in the remapped solution will be assessed with standard error metrics defined as follows.

Note that in order to eliminate potential aliasing errors, the normalization factors

Preservation of the solution gradients in addition to other critical properties, such as local conservation in the remapping procedure, requires

Then, in order to measure accuracy of the solution and its gradient, we introduce two specific global metrics:

In this study,

Global conservation is trivially assessed by evaluating the change in the global integral of the scalar field value on the source mesh and the projected field on the target mesh. We use the following metric to quantify global conservation.

However, we note that this definition for

Global extrema preservation can be assessed via the standard

The error measures

Local extrema preservation can be assessed using a localized difference; i.e., to what degree does the remapped grid cell value fall within the range of surrounding grid cells sampled on the target grid? This consideration motivates us to define a localized difference in extrema:

Note that the definition of the localized differences shown in Eqs. (

For all remapping algorithms evaluated in this comparison study, we conduct iterative two-way (

The workflow necessary to evaluate a given remapping method comprises five consecutive steps described below.

Generate a series of meshes of different topologies and resolutions. We use the cubed-sphere (CS), quasi-uniform Voronoi (MPAS), regular latitude–longitude (RLL), and regionally refined (RMM) grids of the CS and/or MPAS types of varying resolutions to devise the test cases. See Fig.

Given a collection of meshes as in step 1 above, a Python module called

A second Python module called

We emphasize that any existing mesh (such as Yin–Yang

All remapping algorithms evaluated in this study use the mesh data, depending on the scheme, and initial reference solutions on

The final Python module in the metrics suite,

The schematic shown in Fig.

The MIRA workflow for generating the remapping metrics for the intercomparison study.

In the remapping intercomparison study, we consider five scalar test variables defined on the sphere as reference solutions fields. These fields are chosen such that different aspects of the remapper can be evaluated uniformly. Details about the analytical and real-world fields as well as the sampling methodology used in the Python implementation are provided below.

Idealized fields used in this study mirror the approach of

The first analytical field (

Following

These fields are used to test performance for a smooth, well-resolved field and a slightly high-frequency, weakly resolved field with rapidly changing gradients. Given that the analytical expressions for these fields are trivial to evaluate, we can compute the exact numerical errors introduced by the remapping schemes when projecting the fields from

Note that the

Contour plots of analytical fields used in this study:

We also test the performance of each remap technique by regridding real data fields obtained from freely available composite satellite observations. The fields chosen are total precipitable water (

Given the 1D averaged spatial amplitude spectrum for each set of composite satellite data as shown in Fig.

1D averaged spatial amplitude spectrum for real data fields based on composite satellite observations. Fit coefficients over two branches (dotted blue and green lines) correspond to a function

Randomized reconstruction of the real-world fields used in this study.

Global composite data for

Global composite data for

Global topography data are taken from the ETOPO1 Global Relief Model (

The raw satellite data snapshot of

Comparing different remapping algorithms under a unified infrastructure for test problems and metrics collection is a nontrivial task. The metrics defined in this study and the implementation of the various field samplings on arbitrary unstructured meshes have provided large output datasets to analyze the key properties of the remappers under consideration.

Specifically, for uniformly refined experiments, a series of different mesh types (CS, MPAS, RLL)

Details on the number of elements (

Using the field definitions (

The volume of consolidated output metrics data is enormous from this experiment, since 1000 remapping iterations were performed on

Detailed results from the intercomparison study and discussion on the implication of each metric to the remapping scheme are presented next.

The consistency of the high-order remap algorithm implementations can be verified by remapping smooth functions and calculating the spatial convergence order of the resultant approximations on the target mesh after repeated remaps. Using the sampled analytical functions described by

In general, from these studies we observed that convergence rates for both low- and high-degree approximations up to

The conservative schemes implemented in the ESMF package

ESMF: convergence rates for the

Note that we have presented the computed convergence rates from the analysis as is, and qualitatively speaking, the values in Table

In order to better understand the relative accuracy of the first- and second-order conservative remapping implementations in ESMF, we performed a comparative analysis for all grid combinations using standard global error norms. These results are shown in Fig.

ESMF comparison of

The conservative high-order linear maps computed by TempestRemap, as shown in Table

Note that even for smooth solutions, the convergence rates observed for

TempestRemap: convergence rates for the

TempestRemap produces conservative solution projections between mesh combinations that can be third-order with

The remapping scheme based on the meshless generalized moving least squares (GMLS) method demonstrates the flexibility to deliver higher-order convergence for scalar field data. The convergence rates computed for the

GMLS and GMLS-CAAS: convergence rates for the

The convergence rates for the nominal GMLS scheme and the GMLS-CAAS remapping method with a post-processing step in Table

Even though the GMLS-CAAS remaps show lower convergence rates, it provides the benefit of making the scheme globally conservative and monotone. The CAAS algorithm requires runtime modification of the projected fields to ensure global and local bounds, and the nonlinear solution-dependent filter can eliminate Gibbs oscillations, providing better stability during remap operation.

The convergence rates for various polynomial degrees of reconstruction

WLS-ENOR: convergence rates for the

We note that the WLS-ENOR algorithm is equipped with an internal nonlinear filtering (or more precisely, mollification) mechanism to detect sharp gradients and discontinuities in order to adaptively choose the weights during the high-order reconstruction process locally. In contrast to the GMLS-CAAS high-order meshless scheme with a post-processing filter that results in a convergence order degradation, the WLS-ENOR scheme remains consistently accurate for smooth functions up to

While high-order convergence rates are achievable for smooth field profiles of

However, it can be observed that higher than first-order schemes tend to show better behavior on all meshes and fields tested. Additionally, the linear maps computed with ESMF underperform the other schemes, as evident from the gradient error norms

Comparing global error measures for the

Note that in this particular comparison, we selected the highest polynomial degree

All mesh-based

The global field integral error indicating a conservation deficit for the

The expected behavior of preserving global integrals in the remapped solutions, as illustrated in Fig.

Global conservation error metric for the

The WLS-ENOR scheme achieves excellent global conservation by applying adaptive quadrature rules to integrate the functions to (nearly) machine precision and redistributing the conservation errors locally near discontinuous regions. The unmodified GMLS scheme is nonconservative without any post-processing and hence is not presented here. However, using the nonlinear CAAS filtering algorithm with the GMLS remapping scheme provides global conservation to user-specified tolerances.

This conservation metric describes whether the remapped solution preserves the global integral over the domain, which is often a desirable constraint for most scalar variables in order to ensure that the total mass and energy in the closed simulation system remain constant. However, for some scalar fields such as sea surface temperature (SST) or TPW, such a conservation constraint may not be strictly mandated, and this enables the use of even the nonconservative GMLS scheme, which has been demonstrated to achieve excellent accuracy.

Departures away from global extrema, i.e., overshoots and undershoots, provide a useful metric to assess the stability of the remap operator under investigation.
A perfectly stable method has a zero value for this metric. The results for the monotonicity metric as a function of the remap iterations are shown in Figs.

The results clearly demonstrate that the mesh-based remapping schemes such as those in ESMF and TempestRemap implementations do not adhere to the global extrema (both maxima and minima) after the first linear remap operator application. The magnitude of this departure is resolution-dependent, but the behavior is consistent in all cases tested. The use of low-order, dissipative linear maps from ESMF shows a slow drift away from global maxima, which are nearly recovered after several remaps (around 700 in Fig.

In contrast, the high-order TempestRemap solutions show a drift away from the bounds in every case. The mildly damped increase in the metric as a function of remap operator application signals the presence of spurious high-frequency modes in the linear map, which is more prevalent as the degree of reconstruction increases. Obtaining high-order remapped solutions in addition to preserving monotonicity in these schemes will require post-processing filters such as CAAS, which can be used to improve the behavior in TempestRemap for such problems of interest.

Compared to the traditional remapping schemes, the meshless GMLS-CAAS and hybrid WLS-ENOR remap schemes maintain the global bounds for most test cases and fields tested in this study.
Augmenting the non-monotone GMLS scheme with the CAAS bounds preservation algorithm shows excellent, stable behavior for all cases. WLS-ENOR shows relatively high compliance with actual bounds because its non-oscillatory weights near discontinuities essentially preserve the monotonicity and convexity of the solutions. However, when the meshes are sufficiently fine, the functions may appear smooth relative to the grid resolution, and in such scenarios, WLS-ENOR would not apply the adaptive weights and the global extremes may oscillate randomly at a magnitude comparable to the local discretization error, as seen in Fig.

Similar to the global bounds metrics that measure solution monotonicity, the local bounds metrics compute the norm of error resulting from the comparison of the reference sampled field data against the projected field in a one-ring local neighborhood. This metric provides insight into the preservation of local field features on repeated remap operator applications. The computed locality metrics for maxima and minima for various mesh resolutions and real-world fields are shown in Figs.

These results showcase the fact that the high-order TempestRemap implementations perform quite well with minimal feature loss for all fields on all meshes.
In contrast, the low-order ESMF linear maps have high dissipation growth that is dependent on both the resolution and remap iterations. For high-resolution meshes, the amplification of the dissipation at every iteration can even be undamped and grows linearly with the ESMF

For the WLS-ENOR scheme, which uses a hybrid strategy to detect local features and adaptive quadrature rules to resolve a high discrepancy in mesh resolutions, its preservation of local bounds exhibits a strong dependence on the spatial resolution of

In the meshless remapping category, the GMLS schemes utilize the CAAS algorithm to clip and conserve fields in a local subset of the neighborhood for each evaluation point. For GMLS-CAAS, local bounds must be given to CAAS, and it is important to consider how they are determined. As GMLS is a meshless solution technique, local bounds for each site of reconstruction are determined by computing the maximum and minimum of the values in the neighborhood (determined meshlessly using a

One of the key outcomes of this analysis indicates that using nonlinear filtering algorithms like CAAS can provide the benefits of property preservation to achieve global conservation and monotonicity constraints. However, these advantages can be offset by the higher local dissipation effects, especially on disparate mesh resolutions of

We restrict the analysis in this study to all the conservative variations of the schemes chosen. As a result, the global conservation metric is truly satisfied for all cases tested with RRMs. The following subsections provide a detailed comparison of the error resolutions as well as global and local bound preservation metrics.

In order to compare the performance of the different remapping schemes on more realistic and complex unstructured RRMs, the global error norms

We note that the global errors with respect to all error norms are considerably smaller in WLS-ENOR and TempestRemap for the smooth (analytical) field variables sampled on the RRM input meshes. Interestingly, the low-order ESMF implementation shows performance comparable to the high-order GMLS-CAAS(

However, it is imperative to note that when transferring field data with significant

Comparison of global error norm metrics for different remap schemes on the finest (r2) CS-MPAS RRM combination, ordered from best to worst in terms of

In order to make a fair selection of ESMF conservative algorithm, the comparison between the

ESMF comparison of

Similar to the uniformly refined cases, the monotonicity metric

The observed trends in the locality metrics computed on RRMs are similar to those exhibited in the uniformly refined mesh experiments presented earlier. Hence, for brevity, only the

It is imperative to recognize that the high-order remapping schemes from TempestRemap and WLS-ENOR continue to generate minimally diffusive projections on target RRMs. This behavior is quite attractive as local features in the fields, even in the presence of strong gradients, are resolved accurately with very low dissipation away from the sampled reference field data. The locally adaptive, discontinuity tracking WLS-ENOR algorithm shows strongly stable behavior for all test cases.

However, the meshless GMLS-CAAS scheme and the low-order ESMF scheme exhibit severe departures away from these local bounds that are consistent with the relatively larger

One key observation from the global monotonicity metrics shown in Fig.

This work provides a foundation for the systematic computation of key metrics of interest in regridding problems and implements the infrastructure to sample analytical and real-world fields as well as to measure the metrics through a flexible Python code. In the current paper, we have presented detailed experiments to compare remapping schemes for FV–FV discretization settings on three specific mesh families (CS, MPAS, RLL) of varying resolutions, both uniform and regionally refined. There are several research directions that will be explored further in order to strengthen and improve the intercomparison protocol and apply it to more realistic climate science problems.

Four key extensions of interest in this context are

complex meshes, including successive

complex discretizations, other than just FV for source and target components;

complex field descriptions, including vector data and preservation of nonlinear correlations in multiple fields; and

computational efficacy, comparing numerical accuracy and algorithmic performance on next-generation architectures.

Further details regarding these research directions are provided in the following subsections.

Uniformly refined meshes provide a simple infrastructure to test the consistency of remapping schemes, and additionally, regionally refined global grids help identify any spatial-scale dependencies in the algorithms. However, it is important to note that the meshes tested do not include any topological holes that are typical in production climate simulations. A direct extension of the current study would be to perform FV–FV remapping studies on realistic meshes, especially with the MPAS ocean polygonal meshes

Several potential issues may arise when remapping fields on these complex topological meshes: for example, dealing with narrow isthmus regions like Panama, which is also at the boundary between ocean basins, or along the coast of Peru where sea level with high-altitude points from the Andes can influence and contaminate remaps to produce biases that appear along these coastal regions. Such regions can yield incorrect field remaps when using simple nearest neighbor maps, high-order reconstructions, or even when computing the overlay mesh, as the numerical tolerances for querying neighborhood data become an important influencing factor in the overall accuracy.

The metrics definitions for standard error norms, global maxima, and global minima introduced in Sect.

Furthermore, another test study of interest would be to develop a sequence of meshes that slowly deform from the original mesh into some intermediary that resolves moving solution fields like tracer transport, eventually returning back to the original mesh. Such a test is referred to as “cyclic-remap” test suites in the ALE community

The current study focused primarily on element-averaged field data typical of FV discretization of models. Although many atmosphere and ocean models define the scalar fields as element-averaged piecewise constant data, it is imperative to extend the above analysis framework to spectral element (SE) data defined on CS meshes so that it can be remapped onto either FV data on MPAS or RLL meshes or SE data on CS meshes with different resolutions. These extensions would allow us to verify the flexibility of the high-order, conservative remapping schemes tested in the current study under discretization specifications commonly used in climate components like HOMME

During the remapping of climate data in production runs and scientific analysis, it is important to preserve not only scalar fields but also vector fields and derived properties that provide better insights from simulations. For example, to understand the atmospheric flows, analysis of global wind patterns is often performed in weather prediction, which requires the evaluation of derivatives or integrals of the vector wind velocity fields

Typically, such vector fields are treated as collections of unrelated scalar fields that are remapped independently.
However, such approaches are deficient in preserving divergence-free conditions and can be inconsistent, since the propagation of remapping error in the components is not correlated. Care should also be taken to ensure that the regridding of vector components uses a proper 3D Cartesian coordinate system instead of spherical mesh projections, which will not yield consistent vector fields on

Remapping algorithms based on mimetic schemes

The additional characteristic dimension that is essential to recognize in intercomparison studies is that the overall cost to obtain the remaps over the simulation cycle includes both a one-time setup cost and a constantly growing computational load at every coupling step when field data need to be transferred between components. Hence, in order to better describe the computational cost of the remapping algorithms, we could split the effort into an offline cost and an online cost. Note that, often, what is sacrificed in terms of computational performance is gained in the quality of the remapped solution through higher accuracy, global bounds preservation, and strong local feature resolution with minimal dissipation. Since the numerical efficiency and time to compute the solution are competing factors, the usage of advanced remapping algorithms for realistic cases will require a more detailed analysis of the computational complexity at scale. An exception to the performance model derived here is that for

The traditional mesh-based conservative remapping algorithms used in ESMF and TempestRemap require computation of an intersection or supermesh,

Comparison of source field reconstructions showing DoF coupling in linear maps when utilizing different remapping algorithms and degrees of expansion.

Computation for the GMLS-CAAS mesh-free approach on a manifold is dominated by the internal QR with pivoting factorization, requiring

In terms of the computational cost, similar to TempestRemap and GMLS, the transfer for smooth function and the detection of discontinuities in WLS-ENOR can take advantage of some pre-processing steps to build matrix-based operators. Hence, they primarily involve SpMV as the primary online cost as well. Additionally, the resolution of discontinuities requires constructing and solving the generalized Vandermonde systems for each target cell near discontinuities as described in Sect.

While the theoretical complexity requirements for the remapping algorithms under consideration can be considerably different, it is imperative to measure the performance of these implementations on various architectures on standard test problems to gauge overall efficacy (accuracy vs. total computational time for

Remapping algorithms are critically important in climate science applications to maintain key numerical properties during the transformation and transfer of coupled field data between component models. Inaccurate, nonconservative, or highly dissipative remapping operators can introduce numerical artifacts in the coupled fields, which can destroy the high-order accuracy of component solvers and propagate errors into the global nonlinear system representing the climate system. Hence, understanding the behavior of remapping algorithms for Earth system modeling requires standardized numerical definitions that provide better insight into the accuracy of transferred fields, along with the ability to preserve global and local solution bounds to avoid numerically induced instabilities in the system.

In this paper, with these motivations in mind, four different remapping schemes were selected based on the current availability of the software implementations, the maturity of the underlying numerics, and potential computational efficiency that could be obtained for real-world scenarios in comparison to existing state-of-the-art remapping algorithms used in production climate simulations. The mesh-based implementations in the libraries ESMF

Comparing these four distinct remapping algorithms requires a uniform test infrastructure, which provides the framework to create new verification studies and analyze the metrics obtained from the remapping algorithms. To enable such unbiased comparative studies, we introduced several remapping metrics that represent the key properties of remapping algorithms. These include global error measures under various norms like

Furthermore, a flexible workflow built on several Python-based drivers to generate the unstructured meshes of different element topologies and resolutions, including regionally refined meshes, and to accurately sample five element-averaged fields using SPH expansions was provided. With these input meshes, the four remapping algorithms were applied for both the smooth analytical fields and representative real fields in an iterative fashion to compute cyclic projections (

Code and data availability.

The results compiled from various test problems demonstrate that the conservative remapping implementations in ESMF are all first-order accurate for smooth problems, even though the

On the other hand, the hybrid mesh-based and meshless schemes achieve very high order consistently (up to

It is essential to note that while the fully mesh-based remapping algorithms are, in general, insensitive to mesh resolutions or the topology, both the hybrid and meshless schemes are susceptible to larger mesh-dependent dissipation. This is especially evident when the source and target mesh resolutions differ drastically. The use of the CAAS algorithm, when combined with even nonconservative schemes such as GMLS, can provide global conservation and bounds preservation at the cost of added dispersion to control numerical oscillations. In contrast, the built-in discontinuity indicators used by the WLS-ENOR algorithm demonstrate good feature-resolving properties for real fields in all experiments conducted.

These experiments conducted on both uniform-resolution meshes and regionally refined meshes provide valuable insight into the properties of remapping algorithms and their numerical behavior. However, practical use of these algorithms in real-world scenarios requires deeper investigation using more topologically diverse, complex meshes and discretization specifications that are more representative of components used in E3SM and CESM.

Finally, we want to emphasize that the MIRA infrastructure presented in this paper is freely available as an open-source package

Information on the availability of source code for the remapping metrics intercomparison infrastructure featured in this paper, all relevant input meshes, and the final consolidated metrics data for schemes are provided in Table 8.

VSM, JEG, XJ, and PK wrote the paper (with several helpful review comments from PB, PU, and RJ). DM contributed text to the TempestRemap section. JG wrote significant portions of the remapping intercomparison code

At least one of the (co-)authors is a member of the editorial board of

Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We gratefully acknowledge the computing resources provided on Bebop, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-06CH11357, to generate the output for test problems using ESMF and TempestRemap remapping software libraries. We also gratefully acknowledge use of the Common Engineering Environment (CEE) compute servers at Sandia National Laboratories to generate the output for test problems using GMLS. Computational results for WLS-ENOR were obtained using the Seawulf cluster at the Institute for Advanced Computational Science of Stony Brook University, which was partially funded by Empire State Development grant NYS no. 28451. The authors would also like to thank the two anonymous reviewers for their detailed comments, which improved the clarity of a few sections in this paper.

This research is supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (grant no. KJ0403000) as well as the Office of Biological and Environmental Research (grant no. KP1703020) Scientific Discovery through Advanced Computing (SciDAC) program. Additional support was provided by the NOAA Office of Oceanic and Atmospheric Research under the NOAA–University of Oklahoma cooperative agreement (NA11OAR4320072; U.S. Department of Commerce).

This paper was edited by Simone Marras and reviewed by two anonymous referees.