Editorial Type:
Article
Article Type:
Research Article

Arbitrary-Order Conservative and Consistent Remapping and a Theory of Linear Maps: Part II

Paul A. Ullrich Department of Land, Air, and Water Resources, University of California, Davis, Davis, California

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Dharshi Devendran Lawrence Berkeley National Laboratory, Berkeley, California

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Hans Johansen Lawrence Berkeley National Laboratory, Berkeley, California

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Print Publication:
01 Apr 2016
DOI:
https://doi.org/10.1175/MWR-D-15-0301.1
Page(s):
1529–1549
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Abstract

This paper extends on the first part of this series by describing four examples of 2D linear maps that can be constructed in accordance with the theory of the earlier work. The focus is again on spherical geometry, although these techniques can be readily extended to arbitrary manifolds. The four maps include conservative, consistent, and (optionally) monotone linear 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. Arbitrary order of accuracy is supported for each of the described nonmonotone maps.

Corresponding author address: Paul Ullrich, Department of Land, Air and Water Resources, University of California, Davis, 1 Shields Ave., Davis, CA 95616. E-mail: paullrich@ucdavis.edu

Abstract

This paper extends on the first part of this series by describing four examples of 2D linear maps that can be constructed in accordance with the theory of the earlier work. The focus is again on spherical geometry, although these techniques can be readily extended to arbitrary manifolds. The four maps include conservative, consistent, and (optionally) monotone linear 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. Arbitrary order of accuracy is supported for each of the described nonmonotone maps.

Corresponding author address: Paul Ullrich, Department of Land, Air and Water Resources, University of California, Davis, 1 Shields Ave., Davis, CA 95616. E-mail: paullrich@ucdavis.edu
Keywords: Observational techniques and algorithms; Algorithms; Data processing; Mathematical and statistical techniques; Interpolation schemes; Models and modeling; Interpolation schemes; Numerical analysis/modeling

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.

The mapping operator is built so as to satisfy the linear remapping problem: given source mesh , target mesh , and vectorized source mesh density field , define a matrix operator so that
e1
is an accurate representation of the vectorized density field on the target mesh. The first paper in this series described the mathematical properties of the linear mapping operator that were required for conservation, consistency, and (optionally) monotonicity. It further provided an example of how one could use these properties to construct an arbitrary-order linear map from a finite-element mesh to finite-volume mesh.

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).

Fig. 1.
Fig. 1.

A depiction of the four meshes studied in this manuscript: (a) cubed-sphere, (b) great-circle latitude–longitude, (c) diamond mesh, and (d) icosahedral kite mesh.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

Following Ullrich and Taylor (2015), standard error measures are employed:
e2
e3
e4
Here denotes the linear mapping operator; and are discretization operators that take the continuous field ψ to the source and target mesh, respectively; and is an integration operator over the target mesh. Validation of the interpolation methodology again uses the three standard fields described in Ullrich and Taylor (2015), including a smoothly varying function , a rapidly varying spherical harmonic , and an artificial vortex.
Throughout this paper geometric consistency is assumed (Ullrich and Taylor 2015, definition 8). Specifically, this property requires that for each finite element , the sum of all local weights is consistent with the geometric area. For a discontinuous finite element on the source mesh this requirement can be written as
e5
where denotes the weight of degree of freedom k (typically sampled pointwise) and denotes the geometric area of the degree of freedom 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

The finite-volume reconstruction procedure follows Jalali and Ollivier-Gooch (2013), among others. Consider an arbitrary 2D polygonal face on the source mesh () defined by 3D corner points , where . Corner points are connected by great circle arcs in counterclockwise order. A polynomial reconstruction is defined via
e6
where α and β are defined implicitly via the unique solution of
e7
and is the approximate centroid:
e8
That is, α and β represent the normalized distance along the vector connecting the approximate centroid to and , respectively, whereas γ is normalized distance perpendicular to both and (Fig. 2). This third distance measure is necessary due to the potential curvature of the volumes in 3D and allows for the inversion of the linear system (i). The polynomial reconstruction in (6) can be truncated as desired, depending on the preferred character of the reconstruction. We denote the number of coefficients in the truncation by . Two popular truncations of order are triangular truncation, defined by , and rectangular truncation, defined by . In particular, triangular truncation neglects the tensor product terms in the polynomial expansion that have combined exponents above . In our experiments, rectangular truncation appears to produce better quality maps when paired with least squares reconstruction, and so it will be employed in the remainder of this manuscript.
Fig. 2.
Fig. 2.

A depiction of the coordinate system used for defining a polynomial reconstruction over a curved quadrilateral (k = 1, 2, 3, and 4) with centroid . A 2D plane (thin dashed line) is constructed with a basis and . The orthogonal coordinate system is completed with the plane-perpendicular vector . The point x is then written as the linear combination .

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

The polynomial reconstruction in (6) can also be written as the inner product of a position vector , which is composed of some arrangement of the terms , and a vector , composed of the associated reconstruction coefficients (described in section 3e). The expansion in (6) then takes the following form:
e9
For simplicity, the remainder of this text will assume that the first element of corresponds to the constant mode ().

b. Construction of the submap

In Ullrich and Taylor (2015), submaps were defined as linear operators that map a limited set of degrees of freedom from the source mesh to the target mesh. For FV to FV remapping, the submap is constructed for each finite volume and composed via Ullrich and Taylor (2015) in Theorem 1. Construction follows a two stage procedure. First, a fit operator is constructed that maps values of the density variable in faces adjacent to to the coefficients of a polynomial expansion. Second, an integration operator is constructed that maps from the polynomial coefficients to an integrated mass on the target grid. The submap is then expressed as
e10
where is the diagonal matrix whose entries are given by
e11
In this case, is simply the geometric overlap area between source volume i and target volume j (i.e., ).

c. Building the integration operator

The integration operator is composed of rows , which represent integration over target volume of the reconstruction. Since exact integration may be unavailable, quadrature over triangles is used to define the integration operator, as follows. Each overlap region is decomposed into disjoint triangles in accordance with Ullrich and Taylor (2015) (see their section 3). The set of corner points of each triangular region is denoted by , and the area of the triangular region is denoted by . The integration operator over each overlap region is then constructed by using a triangular quadrature rule with to integrate over the polynomial reconstruction:
e12
e13
Integration and averaging over , which will be necessary for verifying conservation, is performed via summation over all target elements, and denoted by
e14
With this definition, the following result holds.

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:
e15
the requirement that and the requirement that the quadrature rule must satisfy
e16
[The black square symbol indicates completion of a proof.]

d. Building the set of adjacent faces

Define as the set of faces that are “adjacent” to in some sense. Given a minimum size for , this set is built as follows:

BuildAdjacentFaceSet(Face f, Integer min_size)

AdjSet ← f

while |AdjSet| < min_size

add all edge neighbors of AdjSet faces to AdjSet

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

There are two key properties that the fit operator must satisfy so that conservation and consistency are ensured. First, for conservation the fit operator must satisfy
e17
that is, the average of the reconstruction over the source element must always yield its own density. For consistency, the fit operator must also satisfy
e18
that is, the fit operator must produce a constant reconstruction when fed the constant field. This claim is proven in the following theorem.

Theorem 1: If satisfies (17) and (18), the linear submap is conservative in and consistent in .

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 , we have
eq1
where is the Krönecker delta.
To show consistency [using (18)]:
eq2
then using (12) and :
eq3
We now describe a technique for constructing the fit operator in terms of a weighted pseudoinverse (Lashley 2002; Weller et al. 2009; Skamarock and Gassmann 2011; Thuburn et al. 2014). Define the density vector as the vector of densities associated with the set . A polynomial reconstruction is defined in with coefficients . The operator then denotes some approximate integration operator that maps the coefficients of the polynomial expansion to the discrete density over :
e19
Note that equality will only hold if , which is generally not the case. For consistency with the integration operator in the source element, we require that satisfy
e20
where the subscript 1,: denotes the complete first row.The remaining components of , which represent the face-averaged integrals of the reconstruction over all adjacent elements, can be determined via any sufficiently high-order quadrature rule. For simplicity, we break up each adjacent element into triangular elements and use an integration procedure analogous to (12).
An equivalent weighted system to (19) can be computed by left-multiplying both sides of this equality by a weighting matrix :
e21
The purpose of the weighting matrix is to reduce the penalty associated with a mismatch between the polynomial reconstruction and the density for faces farther away from . Many choices of are available, although we have had empirical success with the choice
e22
where is the vector of graph distance away from the source element (so the source element has value zero, its edge neighbors have value 1, and so on).

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 Δx, and consider the limit of .

Theorem 2: The weighted Moore–Penrose pseudoinverse applied to densities yields an order reconstruction about when the mesh structure is preserved under refinement.

Proof: By properties of the pseudoinverse,
e23
Consequently, for any polynomial up to degree , the operator will yield the exact polynomial coefficients. To complete the proof, we must now demonstrate that for any field with the reconstruction is . Let denote the total polynomial order of , that is,
e24
Since entries of are integrals of , the kth column of this operator must be . For (23) to be satisfied it follows that the kth row of must then be . By construction, the densities of this field and so Hence, the composed reconstruction must satisfy . ■
As a consequence of Theorem 2, it is clear that is a high-order accurate approximation to the fit operator. However, it can be readily demonstrated that this quantity does not lead to a conservative linear map:
e25
Consequently, we define the corrected fit operator as follows:
e26
where the subscript 2:Nc denotes rows 2 through Nc. It can then be shown that this operator satisfies all desired properties:

Theorem 3: The corrected fit operator in (26) produces a conservative, consistent, and order accurate linear map.

Proof: By the definition of the fit operator and Lemma 1, it follows that satisfies (17). We now show consistency of : for rows m > 1, since and satisfies (23), we have
e27
For the first row, we have
eq4
Combining these results, it follows that satisfies (18). Hence, by Theorem 1 the composed linear map is conservative and consistent.
To show that the corrected operator retains order accuracy, we first observe that the reconstruction coefficients associated with the nonconstant mode are all identical to the uncorrected pseudoinverse, and hence retain the accuracy of that operation. For the constant mode, we are interested in computing the difference between the corrected and uncorrected fit operators:
e28
Right multiplying this difference by and using (20) and Lemma 1, one obtains
e29
This result implies that the correction to the first row does not lie in the polynomial space associated with , and so must be . ■

f. Monotonicity

Monotonicity for the FV to FV remapping operator is guaranteed if and only if . In this case, the global linear remapping operator can be written directly as
e30
This operator simply assumes that the density variable is constant within each source mesh region, and that the amount of mass distributed to each target region is proportional to the overlap area. Conservation, consistency, and monotonicity are trivially demonstrated in this case (Grandy 1999).

g. Results

Standard error norms for finite-volume remapping from cubed-sphere meshes of resolution ne = 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 Np = 1, 2, 3, and 4 are given in Fig. 3. Order convergence in the error is mostly observed for all three fields, except for the smooth field at highest resolution and order of accuracy. In this case it appears that the falloff is due to ill conditioning, likely from the underlying geometry (this effect appears to be consistent across all of the mapping schemes tested). Errors appear to be evenly distributed for the smooth field (not shown) and do not accumulate at the poles as one might expect. Nonetheless, for the relatively rough fields and vortex, convergence rates are as expected. Absolute and error norms are reported for this test in Fig. 4. Increased resolution appears to generally reduce these errors, but clearly not as consistently as with the standard error norms. Consistently monotone behavior is only observed with np1, as expected. Also, these errors appear to significantly diminish when going from a linear (np2) to a quadratic (np3) reconstruction, particularly for and vortex tests. The spatial character of the errors for the vortex test is depicted in Fig. 5, which emphasizes that errors are correlated with roughness in the source field. It is apparent that errors are associated with subgrid-scale variability, which, in turn, drives grid imprinting from the source mesh; and thus errors are significantly reduced when increasing the order-of-accuracy of the reconstruction.

Fig. 3.
Fig. 3.

Standard (left to right) , , and error norms reported for conservative and consistent remapping of (top to bottom) the three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions ne = 15, 30, and 60; rectangular truncation; and for four orders of accuracy Np = 1, 2, 3, and 4.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

Fig. 4.
Fig. 4.

Absolute (a) and (b) error norms reported for conservative and consistent remapping of the (top to bottom) three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions ne = 15, 30, and 60; rectangular truncation; and for three orders of accuracy Np = 2, 3, and 4. (left) Undershoots and (right) overshoots are indicated by circled data points.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

Fig. 5.
Fig. 5.

Spatial distribution of errors for the vortex field for conservative and consistent remapping of the three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions (left to right) ne = 15, 30, and 60; rectangular truncation; and for (top to bottom) three orders of accuracy Np = 2, 3, and 4. Since the errors are approximately symmetric with respect to the vortex, only part of the Southern Hemisphere is plotted.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

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 and note that the procedure for constructing a map for continuous finite elements is analogous, except with a final application of a direct stiffness summation or averaging procedure.

As noted in Ullrich and Taylor (2015), calculation of is difficult and relies on the fact that
e31
where is typically a nonlinear test function associated with degree of freedom i, and the numerical integral is subject to effectively arbitrary underlying geometry. Hence, (31) may not hold in practice (particularly if is evaluated using the pointwise determinant of the metric, rather than via an integration procedure). However, as long as the GLL finite element is geometrically consistent, it is nonetheless possible to construct a conservative, consistent, and monotone linear map. The procedure described here builds the map without the need for constructing explicitly.

a. Building the integration operator

The “first guess” integration operator is defined analogous to (12), except augmented with :
e32
The source grid average is defined analogous to (14), except using (32) for . Since the represent a partition of unity it is also easy to show that Lemma 1 holds for this integrator.
Define as the matrix where row i consists of (this represents the distribution of mass from to all target elements). The composed global map then takes the following form:
e33
Although it can be readily shown that (33) is conservative, it is not consistent since the target grid weight is not determined by the inexact integration procedure inherited from the integration operator:
eq5
Here the mismatch is typically given by the minimum of the GLL quadrature order and the triangular quadrature order. To build a consistent map, we must modify the averaging operator to redistribute the integrated mass within the target element.
For each disjoint finite-element (with ) define overlap regions between the finite-volume mesh and finite-element B as and the set of source volumes that overlap B as . Then define a modified set of integration and averaging operators, denoted by , via
e34
and as the solution of the least squares problem:
e35
e36
This procedure defines minimization problems in free variables with constraints (one constraint is unnecessary due to a linear dependency). Note that for 2D GLL finite elements of order we have . This minimization problem can be trivially transformed into the minimization problem solved in Ullrich and Taylor (2015).
With the modified integration matrix, the composed linear map takes the following form:
e37
Note that the modified integration operators retain the same source grid average as :
Lemma 2: The modified operators satisfy
e38
Proof: For columns m > 1 the result follows immediately from (34) and the definition of . For column m = 1 we have from (36),
e39
which satisfies the lemma since . ■

The key result of this section then follows:

Theorem 5: The linear map , as defined by (37), is conservative and consistent.

Proof: Conservation and consistency for linear maps are determined by Ullrich and Taylor’s (2015) Proposition 1 and 2. Conservation follows from Lemma 2 [using (37) and (17)]:
eq6
And consistency from the definition of [using (37), (18), and (36)]:
e40

b. Monotonicity

The linear map in (37) can be rendered monotone by choosing a piecewise constant reconstruction on the FV mesh () and by leveraging the strategy of Ullrich and Taylor (2015) to remove negative coefficients from . In practice, this option tends to underperform the volumetric strategy discussed in the following section, and so is not analyzed in this paper.

c. Results

Standard error norms are reported in Fig. 6 for finite-volume to finite-element remapping from cubed-sphere meshes of resolution ne = 15, 30, and 60 (1350, 5400, and 21 600 volumes) to a diamond mesh (10 800 elements) with rectangular truncation and three orders of accuracy and on the target mesh. Convergence order is between and in each norm, where error norms are again observed to level off at the highest order and for the smoothest field. Again, for the relatively rough fields and vortex, convergence rates are as expected, even tending toward order convergence. Absolute and error norms are reported for this test in Fig. 7. The behavior of these norms is analogous to that of the finite-volume maps. Monotonicity is not generally expected in this case, even for np1, since the piecewise constant field is being mapped onto a fourth-order basis function that falls out of the range [0, 1]. These errors again appear to improve greatly when going from a linear (np2) to a quadratic (np3) reconstruction, particularly for and vortex tests. The spatial character of the errors for the vortex test are visually identical to the results from Fig. 5 (and so are not reproduced), suggesting that the subgrid reconstruction on the source mesh is the primary driver for these errors.

Fig. 6.
Fig. 6.

As in Fig. 3, but from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh and for four orders of accuracy on the source mesh and on the target mesh.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

Fig. 7.
Fig. 7.

As in Fig, 4, but from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh and for four orders of accuracy on the source mesh and on the target mesh.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

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 will not generally agree with the local weights , and so cannot be used directly to obtain a conservative and consistent map. A depiction of the artificial CVs is given in Fig. 8 for fourth-order GLL finite elements.

Fig. 8.
Fig. 8.

(a) Artificial control volumes associated with degrees of freedom in a fourth-order Gauss–Lobatto–Legendre finite element and associated GLL quadrature nodes. (b) Artificial control volumes in the fourth-order Gauss–Lobatto–Legendre reference element and associated GLL quadrature nodes, with coordinate axes α ∈ [−1, 1] and β ∈ [−1, 1].

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

a. Building the linear map

The linear map is defined as
e41
where and are the integration–averaging (section 3c) and fit operators (section 3e) from the FV to FV map formulation. The operator is a redistribution operator that accounts for the fact that and are generally not equal. It is computed as follows.
For each disjoint finite element , define the local redistribution operator as if or and are otherwise determined from the least squares problem:
e42
e43
where is the Krönecker delta. Observe that if for each degree of freedom then the solution is trivially given by the identity operator over B. The total redistribution operator can then be written as
e44
Each is effectively a submap within the finite element from CVs to quadrature points. As a consequence, monotonicity of this map can be enforced following the procedure described in Ullrich and Taylor (2015) (see their section 3e). Note that if the finite volume to CV map is monotonic (guaranteed for ) and the redistribution is monotonic then the composed map will also be monotonic.

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 , as defined by (41), is conservative and consistent.

Proof: To show conservation [using (43), (14), and (17)]:
e45
and to show consistency [using (18), (12), and (43)]:
e46

b. Results

Standard error norms are reported in Fig. 9 for finite-volume to finite-element remapping from cubed-sphere meshes of resolution ne = 15, 30, and 60 (1350, 5400, and 21 600 volumes) to a diamond mesh (10 800 elements) with a first-order (monotone) finite-volume reconstruction () and three orders of accuracy on the target mesh. Convergence order is between 1 and 2 in each norm. Errors are dominated by the quality of the reconstruction on the source grid, and so do not improve with target grid order. Figure 10 depicts and and shows no overshoots or undershoots. The spatial character of the errors for the vortex test are plotted in Fig. 11. As observed in the error norms, there is little improvement in the overall magnitude of errors as the order of accuracy on the target mesh is increased, but instead there is a tendency toward more finescale structure in the error field. Nonetheless, errors are again largely constrained to regions where the source field shows the largest gradients and grid imprinting is minimal.

Fig. 9.
Fig. 9.

As in Fig. 3, but for conservative, consistent, and monotone remapping of the three idealized fields from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

Fig. 10.
Fig. 10.

As in Fig.4, but for conservative, consistent, and monotone remapping from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

Fig. 11.
Fig. 11.

As in Fig. 5, but under volumetric (monotone) mapping from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

6. Finite-element to finite-element remapping

The final procedure discussed in this paper addresses mapping from a finite-element source mesh (with order of accuracy ) to a finite-element target mesh (with order of accuracy ). The conservative map between finite elements is constructed using Galerkin projection, analogous to the procedure described in Farrell (2009); Farrell et al. (2009); Farrell and Maddison (2011). Specifically, we assume that the continuous field can be expanded on the source mesh as
e47
and on the target mesh as
e48
where and denote the basis functions on the source and target mesh. Equating (47) and (48), multiplying through by and integrating over the domain then leads to
e49
So as to avoid inverting a linear system mass lumping is applied (Farrell 2009):
e50
Consequently, the Galerkin expansion implies a linear map of the following form:
e51

a. Building the discrete map

The map in (51) is conservative and consistent for exact integration, but only approximately satisfies these conditions when inexact integration is used. This section is primarily concerned with the case when exact integration is unavailable. To proceed, the integrated overlap area between each source grid element and target basis function is first approximated using inexact triangular quadrature via
e52
which leads to approximate global integrals of the basis functions via
e53
A finite-element to finite-element map is constructed in two stages: first, for each source element a conservative map is constructed that maps the element to degrees of freedom on the target mesh. Second, for each target element an operator is constructed that redistributes mass so that the composed map maintains consistency. The composed map is then expanded as
e54
where denotes the vector of approximate overlap areas associated with and obtained from (52).
The coefficients of a first-guess Gaussian projection map are computed approximately, again using triangular quadrature:
e55
The conservative map is then obtained via the least squares problem:
e56
e57
The redistribution operator is constructed analogous to the procedure in section 5a. For each finite-element , define the local redistribution operator as if or and are otherwise determined from the least squares problem:
e58
e59
Attaining the expected order of accuracy of this approach is reliant on and as being convergent to the quantities associated with the exact Galerkin projection map, which is in turn satisfied up to the order of accuracy of the triangular quadrature rule. Accuracy also requires that and are chosen such that
e60
We now prove the key result for this section:

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

Proof: Denote the global redistribution operator by
e61
To show conservation [using (59) and (57)],
e62
And to show consistency [using (53) and (59)],
e63

b. Results

Standard error norms are reported in Fig. 12 for finite-element remapping from cubed-sphere meshes of resolution ne = 15, 30, and 60 (1350, 5400, and 21 600 volumes) to a icosahedral kite grid (15 360 elements) with three orders of accuracy . Convergence order is between and in each norm. Again we observe a flattening of the error curve at the highest resolution and order of accuracy for the smooth field. Figure 13 depicts and . There is no evidence of overshoots or undershoots for np2, which corresponds to a bilinear reconstruction, but both overshoots and undershoots are observed at higher orders of accuracy. The spatial character of errors for the vortex field is plotted in Fig. 14. At the coarsest resolution and lowest order of accuracy, errors are concentrated in regions where the gradients in the source field are the largest. In accordance with the observed error norms, we see a rapid improvement in the quality of the solution as resolution and order of accuracy are increased.

Fig. 12.
Fig. 12.

As in Fig.3, but for from the cubed-sphere mesh to the ni = 16 icosahedral kite grid and for three orders of accuracy .

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

Fig. 13.
Fig. 13.

As in Fig. 4, but remapping from the cubed-sphere mesh to the ni = 16 icosahedral kite grid.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

Fig. 14.
Fig. 14.

As in Fig. 5, but under mapping from the cubed-sphere mesh to ni = 16 icosahedral kite grid.

Citation: Monthly Weather Review 144, 4; 10.1175/MWR-D-15-0301.1

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.)

REFERENCES

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    • Search Google Scholar
    • Export Citation

  • Farrell, P. E., 2009: Galerkin projection of discrete fields via supermesh construction. Ph.D. thesis, Imperial College London, 178 pp.

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    • Search Google Scholar
    • Export Citation

  • Farrell, P. E., M. Piggott, C. Pain, G. Gorman, and C. Wilson, 2009: Conservative interpolation between unstructured meshes via supermesh construction. Comput. Methods Appl. Mech. Eng., 198, 2632–2642, doi:10.1016/j.cma.2009.03.004.

    • Search Google Scholar
    • Export Citation

  • Garimella, R., M. Kucharik, and M. Shashkov, 2007: An efficient linearity and bound preserving conservative interpolation (remapping) on polyhedral meshes. Comput. Fluids, 36, 224–237, doi:10.1016/j.compfluid.2006.01.014.

    • Search Google Scholar
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    • Search Google Scholar
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  • Godunov, S., 1959: A difference scheme for numerical computation of discontinuous solutions of equations in fluid dynamics. Matematicheskii Sbornik, 47, 271–306.

    • Search Google Scholar
    • Export Citation

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    • Search Google Scholar
    • Export Citation

  • Guba, O., M. A. Taylor, P. A. Ullrich, J. R. Overvelt, and M. N. Levy, 2014: The spectral element method on variable resolution grids: Evaluating grid sensitivity and resolution-aware numerical viscosity. Geosci. Model Dev., 7, 2803–2816, doi:10.5194/gmd-7-2803-2014.

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    • Export Citation

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  • Ringler, T., M. Petersen, R. L. Higdon, D. Jacobsen, P. W. Jones, and M. Maltrud, 2013: A multi-resolution approach to global ocean modeling. Ocean Modell., 69, 211–232, doi:10.1016/j.ocemod.2013.04.010.

    • Search Google Scholar
    • Export Citation

  • Satoh, M., T. Matsuno, H. Tomita, H. Miura, T. Nasuno, and S.-I. Iga, 2008: Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations. J. Comput. Phys., 227, 3486–3514, doi:10.1016/j.jcp.2007.02.006.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., and A. Gassmann, 2011: Conservative transport schemes for spherical geodesic grids: High-order flux operators for ODE-based time integration. Mon. Wea. Rev., 139, 2962–2975, doi:10.1175/MWR-D-10-05056.1.

    • Search Google Scholar
    • Export Citation

  • Taylor, M., J. Edwards, S. Thomas, and R. Nair, 2007: A mass and energy conserving spectral element atmospheric dynamical core on the cubed-sphere grid. J. Phys.: Conf. Series, 78, 012024, doi:10.1088/1742-6596/78/1/012074.

    • Search Google Scholar
    • Export Citation

  • Thuburn, J., C. Cotter, and T. Dubos, 2014: A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: Comparison of hexagonal–icosahedral and cubed-sphere grids. Geosci. Model Dev., 7, 909–929, doi:10.5194/gmd-7-909-2014.

    • Search Google Scholar
    • Export Citation

  • Ullrich, P. A., and M. A. Taylor, 2015: Arbitrary-order conservative and consistent remapping and a theory of linear maps: Part I. Mon. Wea. Rev., 143, 2419–2440, doi:10.1175/MWR-D-14-00343.1.

    • Search Google Scholar
    • Export Citation

  • Ullrich, P. A., P. H. Lauritzen, and C. Jablonowski, 2009: Geometrically exact conservative remapping (GECoRe): Regular latitude–longitude and cubed-sphere grids. Mon. Wea. Rev., 137, 1721–1741, doi:10.1175/2008MWR2817.1.

    • Search Google Scholar
    • Export Citation

  • Weller, H., 2014: Non-orthogonal version of the arbitrary polygonal C-grid and a new diamond grid. Geosci. Model Dev., 7, 779–797, doi:10.5194/gmd-7-779-2014.

    • Search Google Scholar
    • Export Citation

  • Weller, H., H. G. Weller, and A. Fournier, 2009: Voronoi, Delaunay, and block-structured mesh refinement for solution of the shallow-water equations on the sphere. Mon. Wea. Rev., 137, 4208–4224, doi:10.1175/2009MWR2917.1.

    • Search Google Scholar
    • Export Citation
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Cover Monthly Weather Review
Monthly Weather Review

  • Chesshire, G., and W. Henshaw, 1994: A scheme for conservative interpolation on overlapping grids. SIAM J. Sci. Comput., 15, 819–845, doi:10.1137/0915051.

    • Search Google Scholar
    • Export Citation

  • Farrell, P. E., 2009: Galerkin projection of discrete fields via supermesh construction. Ph.D. thesis, Imperial College London, 178 pp.

  • Farrell, P. E., and J. Maddison, 2011: Conservative interpolation between volume meshes by local Galerkin projection. Comput. Methods Appl. Mech. Eng., 200, 89–100, doi:10.1016/j.cma.2010.07.015.

    • Search Google Scholar
    • Export Citation

  • Farrell, P. E., M. Piggott, C. Pain, G. Gorman, and C. Wilson, 2009: Conservative interpolation between unstructured meshes via supermesh construction. Comput. Methods Appl. Mech. Eng., 198, 2632–2642, doi:10.1016/j.cma.2009.03.004.

    • Search Google Scholar
    • Export Citation

  • Garimella, R., M. Kucharik, and M. Shashkov, 2007: An efficient linearity and bound preserving conservative interpolation (remapping) on polyhedral meshes. Comput. Fluids, 36, 224–237, doi:10.1016/j.compfluid.2006.01.014.

    • Search Google Scholar
    • Export Citation

  • Giraldo, F. X., 2001: A spectral element shallow water model on spherical geodesic grids. Int. J. Numer. Methods Fluids, 35, 869–901, doi:10.1002/1097-0363(20010430)35:8<869::AID-FLD116>3.0.CO;2-S.

    • Search Google Scholar
    • Export Citation

  • Godunov, S., 1959: A difference scheme for numerical computation of discontinuous solutions of equations in fluid dynamics. Matematicheskii Sbornik, 47, 271–306.

    • Search Google Scholar
    • Export Citation

  • Grandy, J., 1999: Conservative remapping and region overlays by intersecting arbitrary polyhedra. J. Comput. Phys., 148, 433–466, doi:10.1006/jcph.1998.6125.

    • Search Google Scholar
    • Export Citation

  • Guba, O., M. A. Taylor, P. A. Ullrich, J. R. Overvelt, and M. N. Levy, 2014: The spectral element method on variable resolution grids: Evaluating grid sensitivity and resolution-aware numerical viscosity. Geosci. Model Dev., 7, 2803–2816, doi:10.5194/gmd-7-2803-2014.

    • Search Google Scholar
    • Export Citation

  • Hill, C., C. DeLuca, V. Balaji, M. Suarez, and A. Da Silva, 2004: The architecture of the Earth System Modeling Framework. Comput. Sci. Eng., 6, 18–28, doi:10.1109/MCISE.2004.1255817.

    • Search Google Scholar
    • Export Citation

  • Jalali, A., and C. Ollivier-Gooch, 2013: Higher-order finite volume solution reconstruction on highly anisotropic meshes. AIAA 2013-2565, 21st AIAA Computational Fluid Dynamics Conf., San Diego, CA, AIAA, doi:10.2514/6.2013-2565.

  • Jiao, X., and M. T. Heath, 2004: Common-refinement-based data transfer between non-matching meshes in multiphysics simulations. Int. J. Numer. Methods Eng., 61, 2402–2427, doi:10.1002/nme.1147.

    • Search Google Scholar
    • Export Citation

  • Jones, P. W., 1999: First- and second-order conservative remapping schemes for grids in spherical coordinates. Mon. Wea. Rev., 127, 2204–2210, doi:10.1175/1520-0493(1999)127<2204:FASOCR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Lashley, R. K., 2002: Automatic generation of accurate advection schemes on unstructured grids and their application to meteorological problems. Ph.D. thesis, University of Reading, 223 pp. [Available online at https://www.reading.ac.uk/web/FILES/maths/Rk_lashley.pdf.]

  • Lauritzen, P. H., and R. D. Nair, 2008: Monotone and conservative cascade remapping between spherical grids (CaRS): Regular latitude–longitude and cubed-sphere grids. Mon. Wea. Rev., 136, 1416–1432, doi:10.1175/2007MWR2181.1.

    • Search Google Scholar
    • Export Citation

  • Menon, S., and D. P. Schmidt, 2011: Conservative interpolation on unstructured polyhedral meshes: An extension of the supermesh approach to cell-centered finite-volume variables. Comput. Methods Appl. Mech. Eng., 200, 2797–2804, doi:10.1016/j.cma.2011.04.025.

    • Search Google Scholar
    • Export Citation

  • Ringler, T., M. Petersen, R. L. Higdon, D. Jacobsen, P. W. Jones, and M. Maltrud, 2013: A multi-resolution approach to global ocean modeling. Ocean Modell., 69, 211–232, doi:10.1016/j.ocemod.2013.04.010.

    • Search Google Scholar
    • Export Citation

  • Satoh, M., T. Matsuno, H. Tomita, H. Miura, T. Nasuno, and S.-I. Iga, 2008: Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations. J. Comput. Phys., 227, 3486–3514, doi:10.1016/j.jcp.2007.02.006.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., and A. Gassmann, 2011: Conservative transport schemes for spherical geodesic grids: High-order flux operators for ODE-based time integration. Mon. Wea. Rev., 139, 2962–2975, doi:10.1175/MWR-D-10-05056.1.

    • Search Google Scholar
    • Export Citation

  • Taylor, M., J. Edwards, S. Thomas, and R. Nair, 2007: A mass and energy conserving spectral element atmospheric dynamical core on the cubed-sphere grid. J. Phys.: Conf. Series, 78, 012024, doi:10.1088/1742-6596/78/1/012074.

    • Search Google Scholar
    • Export Citation

  • Thuburn, J., C. Cotter, and T. Dubos, 2014: A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: Comparison of hexagonal–icosahedral and cubed-sphere grids. Geosci. Model Dev., 7, 909–929, doi:10.5194/gmd-7-909-2014.

    • Search Google Scholar
    • Export Citation

  • Ullrich, P. A., and M. A. Taylor, 2015: Arbitrary-order conservative and consistent remapping and a theory of linear maps: Part I. Mon. Wea. Rev., 143, 2419–2440, doi:10.1175/MWR-D-14-00343.1.

    • Search Google Scholar
    • Export Citation

  • Ullrich, P. A., P. H. Lauritzen, and C. Jablonowski, 2009: Geometrically exact conservative remapping (GECoRe): Regular latitude–longitude and cubed-sphere grids. Mon. Wea. Rev., 137, 1721–1741, doi:10.1175/2008MWR2817.1.

    • Search Google Scholar
    • Export Citation

  • Weller, H., 2014: Non-orthogonal version of the arbitrary polygonal C-grid and a new diamond grid. Geosci. Model Dev., 7, 779–797, doi:10.5194/gmd-7-779-2014.

    • Search Google Scholar
    • Export Citation

  • Weller, H., H. G. Weller, and A. Fournier, 2009: Voronoi, Delaunay, and block-structured mesh refinement for solution of the shallow-water equations on the sphere. Mon. Wea. Rev., 137, 4208–4224, doi:10.1175/2009MWR2917.1.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    A depiction of the four meshes studied in this manuscript: (a) cubed-sphere, (b) great-circle latitude–longitude, (c) diamond mesh, and (d) icosahedral kite mesh.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    A depiction of the coordinate system used for defining a polynomial reconstruction over a curved quadrilateral (k = 1, 2, 3, and 4) with centroid . A 2D plane (thin dashed line) is constructed with a basis and . The orthogonal coordinate system is completed with the plane-perpendicular vector . The point x is then written as the linear combination .

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Standard (left to right) , , and error norms reported for conservative and consistent remapping of (top to bottom) the three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions ne = 15, 30, and 60; rectangular truncation; and for four orders of accuracy Np = 1, 2, 3, and 4.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Absolute (a) and (b) error norms reported for conservative and consistent remapping of the (top to bottom) three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions ne = 15, 30, and 60; rectangular truncation; and for three orders of accuracy Np = 2, 3, and 4. (left) Undershoots and (right) overshoots are indicated by circled data points.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Spatial distribution of errors for the vortex field for conservative and consistent remapping of the three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions (left to right) ne = 15, 30, and 60; rectangular truncation; and for (top to bottom) three orders of accuracy Np = 2, 3, and 4. Since the errors are approximately symmetric with respect to the vortex, only part of the Southern Hemisphere is plotted.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    As in Fig. 3, but from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh and for four orders of accuracy on the source mesh and on the target mesh.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    As in Fig, 4, but from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh and for four orders of accuracy on the source mesh and on the target mesh.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    (a) Artificial control volumes associated with degrees of freedom in a fourth-order Gauss–Lobatto–Legendre finite element and associated GLL quadrature nodes. (b) Artificial control volumes in the fourth-order Gauss–Lobatto–Legendre reference element and associated GLL quadrature nodes, with coordinate axes α ∈ [−1, 1] and β ∈ [−1, 1].

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    As in Fig. 3, but for conservative, consistent, and monotone remapping of the three idealized fields from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    As in Fig.4, but for conservative, consistent, and monotone remapping from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    As in Fig. 5, but under volumetric (monotone) mapping from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    As in Fig.3, but for from the cubed-sphere mesh to the ni = 16 icosahedral kite grid and for three orders of accuracy .

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    As in Fig. 4, but remapping from the cubed-sphere mesh to the ni = 16 icosahedral kite grid.

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    As in Fig. 5, but under mapping from the cubed-sphere mesh to ni = 16 icosahedral kite grid.

Fig. 1.

A depiction of the four meshes studied in this manuscript: (a) cubed-sphere, (b) great-circle latitude–longitude, (c) diamond mesh, and (d) icosahedral kite mesh.
Fig. 2.

A depiction of the coordinate system used for defining a polynomial reconstruction over a curved quadrilateral (k = 1, 2, 3, and 4) with centroid . A 2D plane (thin dashed line) is constructed with a basis and . The orthogonal coordinate system is completed with the plane-perpendicular vector . The point x is then written as the linear combination .
Fig. 3.

Standard (left to right) , , and error norms reported for conservative and consistent remapping of (top to bottom) the three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions ne = 15, 30, and 60; rectangular truncation; and for four orders of accuracy Np = 1, 2, 3, and 4.
Fig. 4.

Absolute (a) and (b) error norms reported for conservative and consistent remapping of the (top to bottom) three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions ne = 15, 30, and 60; rectangular truncation; and for three orders of accuracy Np = 2, 3, and 4. (left) Undershoots and (right) overshoots are indicated by circled data points.
Fig. 5.

Spatial distribution of errors for the vortex field for conservative and consistent remapping of the three idealized fields from the cubed-sphere mesh to the 1° great-circle regular latitude–longitude mesh for cubed-sphere resolutions (left to right) ne = 15, 30, and 60; rectangular truncation; and for (top to bottom) three orders of accuracy Np = 2, 3, and 4. Since the errors are approximately symmetric with respect to the vortex, only part of the Southern Hemisphere is plotted.
Fig. 6.

As in Fig. 3, but from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh and for four orders of accuracy on the source mesh and on the target mesh.
Fig. 7.

As in Fig, 4, but from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh and for four orders of accuracy on the source mesh and on the target mesh.
Fig. 8.

(a) Artificial control volumes associated with degrees of freedom in a fourth-order Gauss–Lobatto–Legendre finite element and associated GLL quadrature nodes. (b) Artificial control volumes in the fourth-order Gauss–Lobatto–Legendre reference element and associated GLL quadrature nodes, with coordinate axes α ∈ [−1, 1] and β ∈ [−1, 1].
Fig. 9.

As in Fig. 3, but for conservative, consistent, and monotone remapping of the three idealized fields from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.
Fig. 10.

As in Fig.4, but for conservative, consistent, and monotone remapping from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.
Fig. 11.

As in Fig. 5, but under volumetric (monotone) mapping from the finite-volume cubed-sphere mesh to the ne = 30 diamond mesh.
Fig. 12.

As in Fig.3, but for from the cubed-sphere mesh to the ni = 16 icosahedral kite grid and for three orders of accuracy .
Fig. 13.

As in Fig. 4, but remapping from the cubed-sphere mesh to the ni = 16 icosahedral kite grid.
Fig. 14.

As in Fig. 5, but under mapping from the cubed-sphere mesh to ni = 16 icosahedral kite grid.
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