## 1. Introduction

In the context of global atmospheric modeling, remapping is needed for transferring data between different model components, which may have substantial different underlying grids, and for transferring data to a structured mesh system for postprocessing and analysis. For instance, many modern global atmospheric models have adopted quasi-uniform grids such as the icosahedral (Satoh et al. 2008) or cubed-sphere grid (Taylor et al. 2007), while unstructured meshes have flourished for global oceanic modeling (Ringler et al. 2013). Desirable properties of remapping schemes include consistency (preservation of a constant field), conservation (no change in total mass), monotonicity (no new extrema), and accuracy. Remapping is either online, requiring a specific algorithm to be executed for each source field, or offline, using a set of precomputed mapping weights in conjunction with a computationally efficient sparse matrix multiply operation. The focus on this series is on the generation of accurate, conservative, consistent, and (optionally) monotone linear offline maps. This paper is the second in the series, following Ullrich and Taylor (2015). Note that we will make frequent references to this earlier paper, and so suggest prospective readers have this earlier work available.

This paper extends on this previous work by describing four new examples of techniques for building linear maps. In the process of developing these algorithms, a number of theoretical results are proven to validate that each map satisfies the desired properties of conservation and consistency. Our focus is on general techniques that allow for arbitrary order of accuracy; although, we note that in the context of remapping, linear monotone maps are restricted to be at most first-order accurate, which is provably optimal as a consequence of Godunov’s theorem (Godunov 1959). The use of the overset grid is again key in the development of these new maps [this concept is closely associated with the supermesh of Farrell et al. (2009) and the notion of common refinement from Jiao and Heath (2004)]. It is assumed that the overset mesh is provided, and we refer to either Ullrich and Taylor (2015) or Farrell et al. (2009) for two potential algorithms for its construction.

In section 3, the generation of finite-volume to finite-volume maps using an overset mesh generation technique is discussed. In sections 4 and 5 we present two techniques for the generation of maps from finite volumes to finite elements. Finally, in section 6 we discuss the generation of maps from finite elements to finite elements via Galerkin projection when exact integration is unavailable.

## 2. Preliminaries

The four meshes used in this paper are depicted in Fig. 1. These include (i) the cubed-sphere, (ii) the great-circle regular latitude–longitude meshes, (iii) the diamond mesh, and (iv) the icosahedral kite grid. The first two of these are also used in Ullrich and Taylor (2015). The diamond mesh is generated by inserting nodes at the center of each cubed-sphere face and then building new quadrilateral faces around each edge (Weller 2014). The icosahedral kite grid is generated by regularly subdividing faces of the icosahedron into subtriangles and then further subdividing each triangle into three quadrilaterals (Giraldo 2001). Meshes (i), (iii), and (iv) have been constructed using the SQuadGen spherical quadrilateral mesh generation utility (https://github.com/ClimateGlobalChange/squadgen) (Guba et al. 2014).

*ψ*to the source and target mesh, respectively; and

*k*(typically sampled pointwise) and

*i*.

## 3. Finite-volume to finite-volume remapping

This section focuses on the development of arbitrary-order conservative and consistent linear maps between arbitrary finite-volume (FV) meshes. The basic procedure we propose involves a local reconstruction operation that converts adjacent volume averages into polynomial coefficients, and a second operator that integrates and averages the reconstruction over all target mesh volumes.

Overlapping volumes for FV interpolation have been previously employed by Grandy (1999) in the design of a first-order conservative interpolation scheme. A conservative method using a second-order linear reconstruction was later developed by Garimella et al. (2007). An analogous procedure known as the Galerkin projection (Farrell et al. 2009) was also extended by Menon and Schmidt (2011) to finite-volume meshes, but again was only assessed for a linear reconstruction. In spherical geometry, overlapping volumes were used by Ullrich et al. (2009) for third-order mapping between cubed-sphere and latitude–longitude meshes. Other methods have been developed for spherical geometry that use approximate overlap volumes, such as Jones (1999) and Lauritzen and Nair (2008).

Finite-volume maps have largely been pursued in an “online” sense; namely, in the form of an algorithm that transforms source mesh averages to target mesh averages. Linear maps, which are pursued in this paper, can also be applied in an “offline” sense, where the coefficients of the map are stored as a sparse matrix and applied via a computationally efficient and readily parallelized sparse matrix multiply. Previous work by Chesshire and Henshaw (1994) leveraged certain properties of the coefficients of this linear operator to impose conservation on interpolating fluxes for solving PDEs. Nonetheless, to the best of the authors’ knowledge, this paper is the first to describe techniques for building arbitrary-order conservative and consistent finite-volume maps in arbitrary geometry.

### a. Arbitrary-order polynomial reconstruction on a 2D surface

*α*and

*β*are defined implicitly via the unique solution of

*α*and

*β*represent the normalized distance along the vector connecting the approximate centroid to

*γ*is normalized distance perpendicular to both

### b. Construction of the submap

*i*and target volume

*j*(i.e.,

### c. Building the integration operator

**Lemma 1:** The integration operator in (12) implies that

**Proof:**The result follows from the observation that the overlap regions are a disjoint set of regions that completely cover the source element:

### d. Building the set of adjacent faces

Define

In most cases (and for the experiments performed in this paper), min_size is chosen to be equal to the number of coefficients in the polynomial expansion. However, for certain source grids this can lead to a poorly conditioned inversion problem when constructing the fit operator. In this case, it may be desired to increase the value of min_size as needed.

### e. Building the local fit operator

**Theorem 1:** If

**Proof:**Note that conservative and consistent linear submaps are defined in Ullrich and Taylor’s (2015) Definition 5 and 6. To show conservation: for all

High-order accuracy of the fit operator is now proven when the mesh structure is preserved under refinement. In this case we denote the average distance between grid points as Δ*x*, and consider the limit of

**Theorem 2:** The weighted Moore–Penrose pseudoinverse

**Proof:**By properties of the pseudoinverse,

*k*th column of this operator must be

*k*th row of

*N*denotes rows 2 through

_{c}*N*. It can then be shown that this operator satisfies all desired properties:

_{c}**Theorem 3:** The corrected fit operator in (26) produces a conservative, consistent, and order

**Proof:**By the definition of the fit operator and Lemma 1, it follows that

*m*> 1, since

### f. Monotonicity

### g. Results

Standard error norms for finite-volume remapping from cubed-sphere meshes of resolution *n*_{e} = 15, 30, and 60 (1350, 5400, and 21 600 volumes) to a 1° great-circle regular latitude–longitude mesh (64 800 volumes) with rectangular truncation and four orders of accuracy *N*_{p} = 1, 2, 3, and 4 are given in Fig. 3. Order

## 4. Finite-volume to GLL finite-element remapping

As opposed to the case of a finite-volume target mesh, the integration operator for a finite-element target mesh must couple together each of the degrees of freedom present in a target mesh element. Although mass can be distributed from a finite-volume source region to a finite-element target region relatively easily, one must be careful that mass is distributed to the degrees of freedom within each Gauss–Lobatto–Legendre (GLL) element in a manner that is both consistent and conservative. For simplicity we consider the case of discontinuous GLL finite elements of order

*i*, and the numerical integral is subject to effectively arbitrary underlying geometry. Hence, (31) may not hold in practice (particularly if

### a. Building the integration operator

*i*consists of

*B*as

*B*as

**Lemma 2:**The modified operators

**Proof:**For columns

*m*> 1 the result follows immediately from (34) and the definition of

*m*= 1 we have from (36),

The key result of this section then follows:

**Theorem 5:** The linear map

### b. Monotonicity

The linear map in (37) can be rendered monotone by choosing a piecewise constant reconstruction on the FV mesh (

### c. Results

Standard error norms are reported in Fig. 6 for finite-volume to finite-element remapping from cubed-sphere meshes of resolution *n*_{e} = 15, 30, and 60 (1350, 5400, and 21 600 volumes) to a

## 5. Finite-volume to GLL finite-element remapping (volumetric)

In this section an alternative approach is pursued for monotone remapping from finite volumes to GLL finite elements, similar to an algorithm implemented in the Earth System Modeling Framework (Hill et al. 2004). Under this approach an artificial set of control volumes (CVs) are introduced for each of the degrees of freedom on the finite-element mesh. By treating the CVs as finite volumes, the FV to FV remapping techniques described in section 3 can be directly employed.

The CVs in the reference element are chosen so that the geometric area of each CV equates to the quadrature weight of that node. For example, for fourth-order GLL quadrature with weights (⅙, ⅚, ⅚, ⅙) over the reference element [−1, 1], CV edges are placed at α = (−1, −⅚, 0, ⅚, 1). However, deformation of the mesh due to the unstructured grid and spherical geometry means this correspondence is not maintained away from the reference element. In particular, the CV areas, denoted by

### a. Building the linear map

*B*. The total redistribution operator can then be written as

With the conditions in (43), the composed linear map in (41) is then readily shown to satisfy the conservative and consistency constraints:

**Theorem 6:** The linear map

### b. Results

Standard error norms are reported in Fig. 9 for finite-volume to finite-element remapping from cubed-sphere meshes of resolution *n*_{e} = 15, 30, and 60 (1350, 5400, and 21 600 volumes) to a

## 6. Finite-element to finite-element remapping

### a. Building the discrete map

**Theorem 7:** The linear map, defined by (54) is conservative and consistent.

### b. Results

Standard error norms are reported in Fig. 12 for finite-element remapping from cubed-sphere meshes of resolution *n*_{e} = 15, 30, and 60 (1350, 5400, and 21 600 volumes) to a

## 7. Conclusions

This paper has introduced four new techniques for the generation of conservative and consistent and (optionally) monotone linear maps between fields on unstructured spherical meshes using the theory of Ullrich and Taylor (2015). These techniques support arbitrary order of accuracy for nonmonotone maps, but are restricted to at most first-order accurate when monotonicity is imposed, in accordance with Godunov (1959). These include maps (i) between two finite-volume meshes, (ii) from finite-volume to finite-element meshes using a projection-type approach, (iii) from finite-volume to finite-element meshes using volumetric integration, and (iv) between two finite-element meshes. A theoretical foundation has been provided in each case to demonstrate that these maps satisfy the desired properties. These maps are useful for coupling together model components that are defined using different grid systems or for post-processing of model data. Future work will focus on nonlinear coupling of linear maps to produce high-order accuracy in smooth solution regions and adoption of these techniques in the context of semi-Lagrangian advection.

## Acknowledgments

The authors thank Mark Taylor for spurring on this work and Miranda Mundt for her quality assurance efforts, particularly with the volumetric formulation. The authors would also like to thank Iulian Grindeanu for helpful discussions on the development of these algorithms. This project is funded through the Department of Energy, Office of Science, Division for Advanced Scientific Computing Research and the “Multiscale Methods for Accurate, Efficient, and Scale-Aware Models of the Earth System” program. The software described in this manuscript has been released as part of the Tempest software package, and is available for use under the Lesser GNU Public License (LGPL). (All software can be obtained from GitHub via the following clone URL: https://github.com/ClimateGlobalChange/tempestremap.git.)

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