Editorial Type:
Article
Article Type:
Research Article

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

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

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Mark A. Taylor Sandia National Laboratories, Albuquerque, New Mexico

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Print Publication:
01 Jun 2015
DOI:
https://doi.org/10.1175/MWR-D-14-00343.1
Page(s):
2419–2440
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Abstract

The design of accurate, conservative, consistent, and monotone operators for remapping scalar fields between computational grids on the sphere has been a persistent issue for global modeling groups. This problem is especially pronounced when mapping between distinct discretizations (such as finite volumes or finite elements). To this end, this paper provides a novel unified mathematical framework for the development of linear remapping operators. This framework is then applied in the development of high-order conservative, consistent, and monotone linear remapping operators from a finite-element discretization to a finite-volume discretization. The resulting scheme is evaluated in the context of both idealized and operational simulations and shown to perform well for a variety of problems.

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

The design of accurate, conservative, consistent, and monotone operators for remapping scalar fields between computational grids on the sphere has been a persistent issue for global modeling groups. This problem is especially pronounced when mapping between distinct discretizations (such as finite volumes or finite elements). To this end, this paper provides a novel unified mathematical framework for the development of linear remapping operators. This framework is then applied in the development of high-order conservative, consistent, and monotone linear remapping operators from a finite-element discretization to a finite-volume discretization. The resulting scheme is evaluated in the context of both idealized and operational simulations and shown to perform well for a variety of problems.

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: Climate models; Grid systems; Interpolation schemes; Model initialization; Software

1. Introduction

The unique characteristics of the atmosphere, ocean, and land surface have led the global modeling community to design component models with distinct numerical methods and meshes. Increasingly there has been a further push toward using different numerical meshes for particular physical processes in order to improve accuracy and efficiency of the modeling system. In either case, some mechanism for communication between these meshes is necessary to couple these components and allow for proper accounting of globally conserved quantities. Consequently, the design of conservative, consistent, and monotone remapping operators for translating between different computational grids on the sphere (hereafter referred to as the remapping problem), has been a persistent issue for global modeling groups.

This manuscript is the first in a series describing the new TempestRemap software package for accurate remapping between meshes on the sphere. Remapping operators are usually constructed via a two-stage process: First a search algorithm determines which regions on the source mesh are geometrically “close” to regions on the target mesh. This procedure is performed to ensure that the mapping maintains geometrical locality. Second, a mapping is defined between source regions and target regions that accounts for subgrid-scale variation of the source field (see, e.g., Jones 1999; Margolin and Shashkov 2003; Ullrich et al. 2009; Farrell et al. 2009; Farrell and Maddison 2011; Dong and Wang 2013).

A robust search algorithm can be particularly difficult to define, since special cases such as coincident grid lines, small overlap regions, and the nonlinearity of spherical geometry can quickly lead to conditioning issues. The search problem can be simplified by using dimension splitting to approximate overlap regions (Lauritzen and Nair 2008), restricting the choice of source and target meshes (Ullrich et al. 2009), or by using approximations to grid lines (Jones 1999). The implementation of TempestRemap described in this paper follows the Earth System Modeling Framework (Hill et al. 2004) by restricting the geometry to meshes composed exclusively of regions whose edges consist of great circle arcs, although there are future plans for supporting grid lines of constant latitude.

Nonconservative mapping operators are generally easy to construct using bilinear interpolation (if monotonicity preservation is required) or high-order finite-differencing techniques. However, these operations are not well suited for arbitrary resolution source and target meshes, and when used in conjunction with ad hoc global conservation fixers, they can produce strange nonlocalized behavior. To define a conservative mapping operator, a common approach has been to use the Gauss–Green theorem (Dukowicz and Kodis 1987) to transform area integrals into line integrals around the boundary of the integration region. This approach has been successfully applied for the conservative remapping problem in the Spherical Coordinate Remapping and Interpolation Package (SCRIP; Jones 1999), and was later used by Ullrich et al. (2009) to define a geometrically exact remapping operator between cubed-sphere and regular latitude–longitude meshes. Unfortunately, the use of the Gauss–Green theorem requires that an analytical potential function be found that accounts for the underlying geometry. This is generally only possible for certain simple cases, and is particularly difficult on completely unstructured meshes. Erath et al. (2013) instead proposed using a nonconservative remapping operator defined from inexact area integration via quadrature that was then rescaled to produce a conservative operator. This approach avoided the ill conditioning that arose at higher spatial resolutions from the line-integral approach (Ullrich et al. 2013), but led to a loss of consistency of the remapping operator. To overcome this problem TempestRemap uses a quadrature-based approach to produce a “first guess” operator that is then projected onto the space of conservative and consistent solutions using a novel least squares formulation. The resulting method avoids the need for line integrals and can be used to guarantee conservation and consistency (and, if desired, monotonicity) of the linear map.

The remapping problem is closely connected to conservative advection of scalar fields, using a technique known as semi-Lagrangian advection. This technique is employed by, for instance, the Conservative Semi-Lagrangian Multitracer Transport Scheme (CSLAM; Lauritzen et al. 2010; Ullrich et al. 2013; Erath et al. 2013). By defining the source or target mesh as the location of Lagrangian fluid parcels at two different points in time, conservative remapping can be employed to define a conservative advection operator.

The outline of this paper is as follows. Section 2 describes the mathematical theory underlying linear remapping operators, and how conservation, consistency, and monontonicity can be described in terms of the coefficients of the remapping matrix. One particular example of the construction of an arbitrarily high-order conservative and consistent (and possibly monotone) remapping operator is then pursued in section 3: the map that takes a discrete field from a nodal finite-element mesh to a finite-volume target mesh. The results of testing the resulting algorithm is then presented in section 4 followed by conclusions in section 5. The appendixes provide a simple example of the construction of a linear map and provide additional details on the search algorithm for the overlap grid.

2. Mathematical foundations

Consider some surface , such as the unit sphere. Functions are discretized by sampling ψ at discrete nodes, via pointwise sampling, or over discrete regions, via an area average. The finite set of discrete nodes or regions is then referred to as the degrees of freedom of a discretization. Note that this definition requires that degrees of freedom be associated with the values of ψ, and not with secondary information such as derivatives of ψ or the coefficients of a spectral expansion (unless those coefficients also correspond to point values, such as the case of nodal finite-element methods). Conserved quantities, such as mass, are represented via a local density variable stored at each degree of freedom. The complete set of all discrete density values is denoted by the vector ψ. The operation of discretizing ψ to ψ is denoted by .

In the remapping problem, discretizations are defined for the source and target meshes. Let denote the degrees of freedom on the target mesh, where and is the total number of degrees of freedom. The set of all degrees of freedom is denoted . Each degree of freedom is then associated with a local weight . For finite volumes the local weight would represent the geometric area of the associated region. For nodal finite elements, the local weight represents the value of the global Jacobian, or some global integral of the associated basis function. The local weights then induce an integration operator (or quadrature rule) denoted by and defined as
e1
where denotes the discretization of on the target mesh, with components . On the unit sphere, one would expect that the degrees of freedom would have complete coverage of the surface, that is, if 1 denotes the vector where every entry is 1 then
e2
although this is not necessarily the case in practice. In particular, since integration over finite elements is governed by the truncation error of the underlying reconstruction, one may observe that (2) only holds approximately. Similar quantities are then defined for the source mesh: let denote the degrees of freedom on the source mesh, with and total count , total set , associated weights , and integration operator .
To remap fields from the source mesh to the target mesh, a remapping operator is defined:
e3
where and are discretizations of on the target and source mesh, respectively. Although the remapping operator can be specified arbitrarily, we are motivated to define a remapping operator that is somehow consistent with the geometry of the underlying problem. That is, we expect
e4
where and denote the discretizations of on the target and source mesh, respectively. Equivalence of (3) and (4) is not guaranteed since information is generally lost during a discretization operation. Three desirable properties of the remapping operator are now defined: namely, conservation, consistency, and monotonicity. These properties are defined as follows.
Definition 1: A remapping operator is conservative if the global mass of any field is maintained across the remapping operation:
e5
Definition 2: A remapping operator is consistent if the constant field is maintained across the remapping operation:
e6
Definition 3: A remapping operator is monotone if the remapping operation cannot introduce additional global extrema:
e7

a. Linear remapping operators

This paper focuses on linear remapping operators. That is, where can be written as a matrix-vector multiply operation:
e8
where denotes the coefficients of . In this context, the three properties described above have a clear meaning in terms of the coefficients of .
Proposition 1: The linear remapping operator is conservative if
e9
Proof: From (5) and the definition of the integration operator in (1), conservation can be written as
e10
Then using (8),
e11
However, since (11) must hold for all fields , equivalence implies (9). ▪ [Black square symbol (▪) indicates completion of a proof.]
Proposition 2: The linear remapping operator is consistent if
e12
Proof: From (6) and (8),
e13
Proposition 3: The linear remapping operator is monotone if it is consistent and
e14
Proof: Assume is monotone. The field satisfies and so , , which in turn implies and so consistency is satisfied. To show assume such that . Let and . Then
e15
which contradicts (7).
Now assume is consistent and satisfies (14). Then
e16
where the inequalities hold because of the nonnegativity of and the last equality holds because of consistency. The result is analogous for . ▪
Note that if is conservative and consistent, it also follows that the source and target meshes must have the same area:
e17
This result further implies that for source and target meshes that do not have the same area it is impossible to define a linear remapping operator that is both conservative and consistent.

To clarify the mathematical notation used here, an example linear remapping operator is provided in appendix A.

b. Local conservation

To introduce the concept of local conservation, one needs to first define some notion of geometric locality. Geometric regions associated with degrees of freedom and are denoted as and . The set of all geometric regions on the target and source meshes are denoted as and . The overlap region associated with and is denoted by (see Fig. 1). If then and are said to be local. The set of all overlap regions is referred to as the overlap mesh and denoted by (note that the overlap mesh is sometimes referred to as the supermesh in the literature). Analogous to the source and target meshes, regions on the overlap mesh are associated with corresponding local weights , which must satisfy
e18
The definition of and is sensitive to the choice of discretization (finite volume vs finite element), as follows:

  • For finite-volume discretizations, is subdivided into regions that have a one-to-one correspondence with degrees of freedom by encoding the volume average. Hence, for finite volumes one can say that and are local if there is any geometric overlap between corresponding regions and .

  • For nodal finite-element discretizations, degrees of freedom are encoded as pointwise values, or equivalently as coefficients associated with a particular characteristic function (a modal characterization). An analogous definition of locality to finite volumes is obtained in terms of the support of the characteristic function associated with a particular degree of freedom; that is, and associated with particular degrees of freedom and are the geometric regions where the corresponding characteristic functions are nonzero, along with their closure. Note that finite volumes specialize this definition under the imposition that each region has only one degree of freedom encoded via the constant characteristic function.

Fig. 1.
Fig. 1.

A simplified depiction of a source mesh, target mesh, and overlap mesh along with associated element indices. Region is local with , , , and .

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

Local weights may be calculated via integration over the overlap region,
e19
where and are functions associated with the degrees of freedom and that satisfy
e20
For a finite-volume discretization is equal to 1 within the associated region (and analogously if discretized on the target mesh);whereas for a finite-element discretization is defined by the characteristic function associated with the degree of freedom (more on this in section 3a). Note that numerical errors may make exact computation of (19) difficult without the use of an advanced numerical integration technique, particularly in a manner that is consistent with (18). However, when mapping from a finite-element mesh to a finite-volume mesh this paper will only rely on knowing overlap areas , which can be computed exactly.

The notion of locality then motivates the definition of a locally conservative operator.

Definition 4: The linear remapping operator is locally conservative if it is conservative and
e21

c. Local submaps

The notion of locality is particularly handy when constructing the remapping operators, since the global linear map can be constructed using linear submaps that are only associated with a limited set of degrees of freedom from the source mesh. Local submaps further possess analogs of conservation and consistency that are helpful for building global linear maps.

Definition 5: A linear submap operator is conservative in if
e22
This definition could hold for any set of points A, but typically the set A consists of points that share a common geometric region , such as in the case of a finite-element discretization. For a finite-volume discretization, source elements are usually considered in isolation and so the set A consists of only a single degree of freedom.
Definition 6: A linear submap operator is consistent in if
e23
When constructing linear submaps, the set B is typically the set of degrees of freedom on the target mesh that are local to degrees of freedom A on the source mesh.

Definition 7: A set of linear submaps is complete if (i) , (ii) , (iii) implies , (iv) is conservative in for , and (v) is consistent in for .

These definitions then motivate the following result, which is the fundamental theory for constructing global remapping operators as a combination of local submaps:

Theorem 1: Let be a complete set of linear submaps which are conservative in . Then the global linear map constructed via
e24
is conservative and consistent.
Proof: To show consistency we use property (i), (ii), and (v) (definition 7) and (23):
eq1
Here is an indicator that is 1 if and 0 otherwise.
Conservation follows almost immediately from property (ii), (iii), and (v) (definition 7) and (22), which collectively imply that only one linear submap will have a nonzero weighted column sum:
eq2
Note that if the linear submap is monotone then the global composition will inherit some notion of local monotonicity. Local monotonicity is even stronger than the global monotonicity described in definition 3, in that the global map will not introduce additional local extrema.

3. Remapping finite elements to finite volumes

A global mapping operator from finite elements to finite volumes is now developed using the theory of section 2. Consistent with the notion of degrees of freedom representing the values of , this paper focuses specifically on nodal finite-element methods over the set of Gauss–Lobatto–Legendre (GLL) nodes. The set of degrees of freedom on the source mesh are defined at the GLL nodes within the reference element (see Fig. 2). The operator is first developed for discontinuous finite elements (i.e., admitting collocated degrees of freedom between adjacent elements); if continuous finite elements are used, as in the case of the spectral element method, the rows of the discontinuous remapping operator can be combined via direct stiffness summation (Deville et al. 2002) without affecting conservation, consistency, or monotonicity of the operator. As a consequence of (17), a conservative map from finite elements to finite volumes only exists if the degrees of freedom of the source (finite element) mesh satisfy some notion of geometric consistency.

Fig. 2.
Fig. 2.

(a) Geometric distribution of degrees of freedom in a fourth-order Gauss–Lobatto–Legendre finite element. (b) Degrees of freedom in the fourth-order Gauss–Lobatto–Legendre reference element, with coordinate axes and .

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

Definition 8: A finite element on the source mesh with region and degrees of freedom is geometrically consistent if
e25
This definition implies that the geometric area of the finite-element mesh must be exactly distributed over all degrees of freedom.

Theorem 2: There exists at least one consistent, conservative, monotone linear submap from a geometrically consistent GLL finite element of order to the target mesh.

Proof: By construction. Consider a single GLL element with nodal degrees of freedom, taken from a subset of all degrees of freedom on the source mesh via the indexing function with and . A linear submap is defined by a weighted average over all GLL nodes:
e26
Consistency over is easily demonstrated from (23). Conservation over follows by observing for and
e27
which in turn implies
e28
Monotonicity follows since is consistent and all entries of are nonnegative. ▪

Although the linear submap satisfies conservation, consistency, and monotonicity, it is undesirable as a linear remapping operator since it “averages out” the subgrid-scale variation associated with the finite-element discretization. Consequently the remainder of this section will focus on constructing an improved linear submap operator. To this end, the following basic algorithm is followed:

Given SourceMesh and TargetMesh calculate

 OverlapMesh

For each source region fi in SourceMesh

 Compute a "first guess" conservative and

    consistent linear submap from fi to

   the OverlapMesh

 Project the "first guess" map onto the

   space of exactly conservative and

   consistent linear submaps

 If monotonicity is required, adjust

   coefficients accordingly

 Compose the linear submap in the global

   remapping operator R

Store global remapping operator R

The generation of the overlap mesh follows the algorithm described in appendix B. Since the global map is simply a composition of submaps that are defined over source elements on the finite-element mesh, the remainder of this section will simply focus on construction of submaps with the desired properties.

a. Choice of basis functions

Construction of an approximately conservative map relies on the use of basis functions to provide a continuous analog to the nodal discretization. Two requirements are imposed on these functions. First, the basis functions must also be characteristic functions—that is, at each GLL node exactly one basis function must take the value 1 and all other basis functions must have value 0. Second, for the sake of consistency, the basis functions must be a partition of unity over the finite element. Besides these two requirements, the choice of basis is at the discretion of the user.

For quadrilateral elements we choose basis functions defined via a tensor product, wherein 1D basis functions are cross multiplied to yield 2D basis functions and a corresponding reconstruction:
e29
The coordinates and are defined implicitly via the coordinate transform of Guba et al. (2014). For a quadrilateral region on the source mesh with corner points arranged in counterclockwise order:
e30
Choosing basis functions that are polynomials of maximum degree (the cardinal functions over GLL points and the standard nodal finite-element basis) leads to a non-monotone remapping operator of the highest formal order of accuracy; whereas choosing a set of basis functions with the limited range [0, 1] leads to a low-order, but monotone, operator. Approximate conservation is enforced by choosing a basis whose global integral equals its associated nodal weight. Consistency follows as long as the set of all basis functions is a partition of unity. Herein two choices of basis functions are made.

1) Nonmonotone basis

Our nonmonotone basis over GLL elements is given by the cardinal functions over GLL nodes (Boyd 2001, his appendix F). In terms of the Legendre polynomials of order , denoted , these are
e31
with corresponding weights
e32
The cardinal functions are plotted in Figs. 3a,c for . For 2D GLL finite elements, these cardinal functions and weights are written via tensor product:
e33
Given some local notion of area (such as the Jacobian associated with a coordinate transform), the local weight at the GLL point is typically approximated as the product . However, since this notion of is not geometrically consistent, we instead suggest a closely related, but modified definition .
Fig. 3.
Fig. 3.

(a),(b) Third- and (c),(d) fourth-order GLL basis functions used for the continuous reconstruction: (a),(c) GLL basis and (b),(d) monotone GLL basis.

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

2) Monotone basis

The monotone basis uses a set of monotonized cardinal functions that resemble the standard non-monotone cardinal functions in (31) under the further constraint . However, to enforce conservation we further require
e34
For , actually satisfies these criteria. However, for larger values of it is the case that over some interval, and so the standard cardinal functions do not satisfy the desired monotone property. Consequently a new set of basis functions are constructed that satisfy consistency and conservation but with the limited range [0, 1]. For there is a unique solution for piecewise quadratic polynomials given by
e35
For , there is no solution with piecewise quadratic polynomials. Piecewise cubic polynomials admit one free parameter, which we arbitrarily choose so that the reconstruction is continuous. This choice yields the following solution:
e36
Monotonized cardinal functions can be similarly specified for . These functions are plotted in Figs. 3b,d for .

One curious result emerges from this construction: Although for the monotone basis can correctly capture linear variation over a finite element, for all monotone basis functions have zero derivative at interior GLL nodes, and consequently cannot represent linear variations within the element. This suggests that smoothly varying fields may be more poorly captured when constructing a monotone map with (as we shall see later). The choice of a monotone basis that avoids this problem is confounded by the need for conservation, automatically eliminating the second-order bilinear interpolant as an option.

b. Building a first-guess submap

The first-guess submap required by this algorithm only needs to satisfy the conservation and consistency property approximately. Here it is constructed by using high-order triangular quadrature to integrate each characteristic function over all polygonal regions on the overlap mesh via polygonal subdivision (see Fig. 4). The triangular quadrature rules employed by this package are given by Dunavant (1985), and depicted in Fig. 5 for triangular quadrature rules of order 1, 4, and 8. The triangular quadrature rule should at least match the order of accuracy of the finite-element method to ensure the map exhibits the correct order of accuracy. Only triangular quadrature rules with nonnegative weights are considered, since the use of negative weights could lead to a loss of monotonicity. Given a triangular region with corner points connected via great circle arcs, the quadrature rule is applied via
e37
where is defined in (29). This procedure yields a submap that is consistent as long as the basis functions are a partition of unity, but nonconservative since inexact quadrature is employed.
Fig. 4.
Fig. 4.

Subdivision of (a) a quadrilateral and (b) pentagon into triangles.

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

Fig. 5.
Fig. 5.

(a)–(c) First-, fourth-, and eighth-order triangular quadrature nodes.

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

c. Consistency and conservation enforcement

Since conservation is not guaranteed by the above procedure, we now enforce conservation by orthogonal projection of the submap onto the space of conservative and consistent maps. In some sense, this projected solution is optimal since the projected map is closest (in the sense of Euclidian distance) to the first-guess map. The projection operation is performed by solving a least squares problem for all coefficients in the submap. This highlights a key strength of dividing the global map into submaps; namely, enforcement of global conservation and consistency requires the solution of a large number of small, inexpensive optimization problems (one for each submap), rather than one very expensive global optimization problem.

Given an arbitrary linear submap with source elements and target elements, the corresponding conservative and consistent remapping operator is obtained as follows:
e38
The least squares problem is solved directly via the Lagrangian. Vectors and are defined with elements , , and , , respectively. Note that the result in (17) implies that the consistency and conservation conditions are linearly dependent, and so conservation is only imposed for the first elements. The Lagrangian takes the following form:
e39
The unique minimizer of the Lagrangian is then obtained by differentiating with respect to all coefficients , , and and leads to the linear system:
e40
where is the matrix defined by the derivatives and . This system can be solved efficiently via Schur complement:
e41
Note that to further improve the efficiency of this calculation, the matrix can be specified directly. If we define
e42
then
e43

d. A note about convergence

An advantage of using the least squares procedure is that it does not affect the accuracy (in the sense of convergence) of the first-guess map, in accordance with the following theorem.

Theorem 3: If is constructed using the non-monotone reconstruction in (31) with a triangular quadrature rule of at least order , then is convergent with order .

Sketch of proof: For a sufficiently smooth field define the exact map via
e44
By construction, must be conservative and consistent. Note that since order quadrature is used for constructing , will converge to with order . However, since (38) represents the closest submap (in the sense of Eulerian distance) to that is conservative and consistent, then
e45
and so is also convergent with order . ▪

e. Monotonicity preservation

To impose monotonicity, the least squares problem in (38) can be augmented with an additional boundedness condition given by (14). Solving the resulting constrained and bounded least squares problem can then be done via an interior point method (see, e.g., Boyd and Vandenberghe 2009). However, this additional criteria can be computationally taxing, and so another approach is used in practice. After computing the unique conservative and consistent submap from (38) using the procedure described above, the resulting linear submap may contain small negative values that need to be removed. To do so, the following theorem is used:

Theorem 4: If and are conservative and consistent linear submaps over and , respectively, then for , is a consistent and conservative linear submap.

Proof: By linearity of (22) and (23). ▪

Consequently, if is the conservative and consistent linear submap obtained from the least squares procedure then the linear submap constructed via
e46
is also conservative and consistent. Monotonization of then simply relies on finding a value of ω sufficiently large that has no negative entries. In fact, the choice
e47
meets this criterion and so is used to define a monotone submap.

4. Numerical results

The meshes used for validation of the linear maps are plotted in Fig. 6, although only meshes in Figs. 6a–c will be used for the idealized study. These include (Fig. 6a) an equiangular cubed-sphere mesh, (Fig. 6b) a great circle latitude–longitude mesh, and (Fig. 6c) a geodesic mesh. The great circle latitude–longitude mesh is constructed analogous to a regular latitude–longitude mesh, but all edges are approximated as great circle arcs (note that in a regular latitude–longitude mesh lines of constant latitude are not great circle arcs). This approximation has the greatest deviation from the regular latitude–longitude mesh in the polar region (Ullrich et al. 2009). The geodesic mesh is constructed by taking the dual of an icosahedral mesh, which is in turn obtained by subdividing the triangular faces of an icosahedron into subtriangles. The resulting mesh is composed largely of hexagons, with exactly 12 pentagons appearing due to the icosahedral corner nodes.

Fig. 6.
Fig. 6.

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

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

The analysis mirrors the approach of Lauritzen and Nair (2008): we consider three idealized test cases of varied complexity to understand the error measures produced by the linear maps from GLL elements to finite volumes. The three analytical fields studied are depicted in Fig. 7. Following Jones (1999) and Lauritzen and Nair (2008) the first field is a relatively smooth function resembling a spherical harmonic of order 2 and azimuthal wavenumber 2, given by
e48
The second field is a relatively high-frequency wave similar to a spherical harmonic of order 32 and azimuthal wavenumber 16, given by
e49
These fields are used to test performance for a smooth well-resolved field and a high-frequency poorly resolved field with rapidly changing gradients. The third field is a dual-stationary vortex (Nair and Machenhauer, 2002). The field is given by
e50
where the radius , with angular velocity:
e51
and normalized tangential velocity:
e52
The refer to a rotated spherical coordinate system with a pole located at . Following Lauritzen and Nair (2008) we choose , , , and .
Fig. 7.
Fig. 7.

(a)–(c) Contour plots of the three test fields used in this study. The fields in (a) and (b) take on values in the range [1, 3]. The field in (c) takes on values in the approximate range [0.46, 1.54].

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

Standard error measures are employed:
e53
e54
e55
In some general sense, the error measure identifies errors in large-scale features, identifies errors in small-scale features, and identifies the largest pointwise error. The error measures and identify undershoots and overshoots, respectively, by taking on positive values when the global extreme values are enhanced.

In the following sections the source discretization is generated by sampling ψ at each nodal GLL point on the source mesh. The results of the remapping operation are compared to a target discretization generated via eighth-order triangular quadrature over each polygon on the target mesh. The default configuration further uses a fourth-order triangular quadrature rule to construct the first-guess map.

a. Cubed-sphere mesh to great circle latitude–longitude mesh (nonmonotonic)

Figure 8 shows standard error measures for the conservative and consistent linear map from the cubed-sphere grid with elements per panel () to the great circle latitude–longitude grid with 1° grid spacing (consisting of 360 longitudinal elements and 180 latitudinal elements). The number of GLL nodes per element (and, hence, the order of accuracy of the linear map) is given by , or 4. All results are conservative to machine truncation (not shown). Error measures are smallest for the smooth field, as expected. All fields further show progressively decreasing error norms with increasing and . Further, the and vortex meshes show convergence rates that closely match . The convergence rate for the field is almost , except for the fourth-order scheme, which is not quite fourth-order convergent at the finest resolution. The loss of perfect convergence is due to insufficient resolution on the target grid, which affects both the construction of the first-guess map and the evaluation of the reference solution.

Fig. 8.
Fig. 8.

(left to right) Standard , , 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 latitude–longitude mesh for cubed-sphere resolutions and for three orders of accuracy .

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

Figure 9 shows and for the mapping problem above. In general, there is a consistent decrease in the errors of the extreme values of each field, although these results are far less consistent than the results for the norms. Overshoots and undershoots are observed in many of the simulations with and are identified by circled data points.

Fig. 9.
Fig. 9.

Absolute (a) and (b) 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 latitude–longitude mesh for cubed-sphere resolutions and for three orders of accuracy . Circled data points indicate that the global minimum–maximum has been enhanced (i.e., that monotonicity was not maintained).

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

b. Cubed-sphere mesh to great circle latitude–longitude mesh (monotonic)

Figure 10 provides error norms for the cubed-sphere to great circle latitude–longitude grid mapping described above, except here with strict monotonicity enforced on the linear map via the procedure described in section 3e. The error norms are significantly worse, compounded by the fact that any monotone method is limited to at most second-order convergence. As expected (following the discussion in section 3a), the results are actually worse than the results for the field. Figure 11 shows and for the conservative, consistent, and monotone map. These error norms are always negative, which confirms that the global minimum and maximum are not enhanced by the linear map.

Fig. 10.
Fig. 10.

As in Fig. 8, but for conservative, consistent, and strictly monotonic remapping.

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

Fig. 11.
Fig. 11.

As in Fig. 9, but for conservative, consistent, and strictly monotonic remapping.

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

c. Cubed-sphere mesh to geodesic mesh (nonmonotonic)

To verify robustness of the algorithm, remapping has also been performed from the cubed-sphere grid to the geodesic mesh (generated from triangular elements along each face of the icosahedron). Standard error measures are plotted in Fig. 12 along with / in Fig. 13. The error norms show generally consistent behavior as with the previous study, suggesting that the results are largely independent of the target mesh.

Fig. 12.
Fig. 12.

As in Fig. 8, but for remapping to the geodesic mesh for cubed-sphere resolutions and for three orders of accuracy .

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

Fig. 13.
Fig. 13.

As in Fig. 9, but for remapping the geodesic mesh for cubed-sphere resolutions and for three orders of accuracy .

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

d. Refined cubed-sphere mesh to great circle latitude–longitude mesh (real data)

To test if the algorithm performs well in practice, the software is tested for remeshing of data from a variable-resolution cubed-sphere mesh (Fig. 6d) to the great circle latitude–longitude grid (Fig. 6b). The variable resolution mesh has been designed for a study of California climatology, and so provides an enhancement to 0.25° resolution over California from 1° global resolution. Integration has been performed using the Community Earth System Model spectral element dynamical core (Neale et al. 2010) using the variable-resolution capability described in Zarzycki et al. (2014). The result of the remapping algorithm is plotted in Fig. 14, showing a relatively smooth field (surface pressure) and a highly discontinuous field where monotonicity preservation is necessary (percentage plant functional type). Overall the algorithm performs well, with no obvious grid imprinting apparent on the result.

Fig. 14.
Fig. 14.

Two “real data” tests for remapping from the variable resolution cubed-sphere mesh (Fig. 6d) with to a 0.25° great-circle latitude–longitude grid (Fig. 6b). (a) Surface pressure from a variable resolution simulation using conservative and consistent remapping. Observe that the detail of the result is much finer between 135° and 90°W in the Northern Hemisphere, corresponding to the region of highest mesh refinement. (b) Percentage plant functional type (barren land) using conservative, consistent, and monotone remapping. This field is highly discontinuous and requires that the data are constrained to the interval to be considered meaningful.

Citation: Monthly Weather Review 143, 6; 10.1175/MWR-D-14-00343.1

5. Conclusions

A mathematical theory underlying for conservative and consistent (and optionally monotone) linear maps between meshes on the sphere has been presented. To demonstrate the applicability of this theory, an algorithm has been developed for constructing arbitrary-order conservative and consistent linear maps between finite-element and finite-volume meshes. This method was then tested using a cubed-sphere source mesh and a great circle latitude–longitude target mesh or geodesic target mesh. The resulting remap scheme has been demonstrated to have the correct convergence rate for polynomials up to cubic degree (fourth order), although there is no fundamental limit on the order of the scheme. A technique for constructing conservative, consistent, and monotone maps was also discussed and led to second-order convergent linear maps. Testing was also performed on real data and the results were observed to be satisfactory for real applications.

This algorithm has been extended for generating linear maps from finite volumes to finite volumes and/or finite elements, which will be the topic of a future manuscript. It is also anticipated that the search algorithm and quadrature rule will be extended to support grid lines of constant latitude. This work will be used as a basis for constructing a semi-Lagrangian advection scheme on the sphere, which provides high-order accuracy on arbitrary meshes.

Software availability

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/paullric/tempestremap.git.

Acknowledgments

The authors thank Iulian Grindeanu and Vijay S. Mahadevan for several helpful discussions on the development of the search algorithm. The authors would like to further thank Hans Johansen and Dharshi Devendran for their discussions on the development of the remapping algorithm. We would also like to thank the input of the three anonymous reviewers for improving the clarity of the manuscript. 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.

APPENDIX A

An Example Linear Map

This appendix provides a simple 1D example of a finite-element to finite-volume linear map. Consider the 1D interval covered by one finite element with and divided into three finite volumes of equal width. The regions on the finite-element mesh are regions of support for the nonmonotone basis functions:
ea1
and on the finite-volume mesh are regions of equal width:
ea2
Degrees of freedom on the finite-element mesh are stored at nodal points:
ea3
Local weights on the source mesh are given by (32):
ea4
and on the target mesh by the geometric area:
ea5
Observe that these local weights satisfy (17) and geometric consistency (definition 8), for . By using exact integration over the characteristic functions on this finite element we can then obtain a linear mapping operator:
ea6
Note that since exact integration is used, this operator is already conservative and consistent and so is unaffected by the least squares procedure (section 3c). The operator is not monotone, as is apparent from negative entries , , , and . If an inexact integration procedure were used, the least squares projection could be necessary to enforce conservation.

APPENDIX B

Overlap Mesh Generation Algorithm

This appendix provides an outline of the algorithm used for generating the overlap mesh. For serial overlap mesh generation this approach is far from optimal, but is potentially easier to parallelize than other methods, such as the advancing front method described in Farrell and Maddison (2011). Improved overlap mesh generation remains a topic for future work. The main function simply loops through all faces on the first mesh, first generating a path around the boundary of each face that accounts for intersections with the second mesh, and then follows the path to generate all faces contained within the path. The pseudocode for this algorithm is as follows:

GenerateOverlapMesh ()

 for all faces f in FirstMesh

 OverlapPath p = GenerateOverlapPath (f)

  GenerateOverlapFaces (p, OverlapMesh)

GenerateOverlapPath (FirstFace)

 OverlapPath = {}

 CurrentNode = first node of FirstFace

 CurrentSegment = line connecting    CurrentNode to second node of FirstFace

 Find SecondFace on SecondMesh from  CurrentNode

 for all edges e1 in FirstFace

  while segments still remain in edge

   for all edges e2 of SecondFace

    if CurrentSegment intersects e2

     determine intersection node NextNode

     add new edge [CurrentNode, NextNode]  to OverlapPath

     update CurrentNode, CurrentSegment

   if CurrentSegment does not intersect  any edges of SecondFace

    set next FirstFace edge

    break segment loop

GenerateOverlapFaces (OverlapPath,  OverlapMesh)

 for all remaining edges e in OverlapPath

  remove edge e from OverlapPath

  if e intersects an edge on SecondMesh

   follow SecondMesh edges until reintersect with OverlapPath

  else continue

  if a closed element has been completed

   add a new Face to OverlapMesh

 add all Faces from SecondMesh interior to OverlapPath to OverlapMesh

REFERENCES

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  • Dukowicz, J. K., and J. W. Kodis, 1987: Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations. SIAM J. Sci. Stat. Comput., 8, 305–321, doi:10.1137/0908037.

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    • Search Google Scholar
    • Export Citation

  • Farrell, P., 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
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    • Search Google Scholar
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  • Hill, C., C. DeLuca, 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.

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  • 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.

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  • 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.

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    • Export Citation

  • Lauritzen, P. H., R. D. Nair, and P. A. Ullrich, 2010: A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid. J. Comput. Phys., 229, 1401–1424, doi:10.1016/j.jcp.2009.10.036.

    • Search Google Scholar
    • Export Citation

  • Margolin, L., and M. Shashkov, 2003: Second-order sign-preserving conservative interpolation (remapping) on general grids. J. Comput. Phys., 184, 266–298, doi:10.1016/S0021-9991(02)00033-5.

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    • Export Citation

  • Nair, R. D., and B. Machenhauer, 2002: The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere. Mon. Wea. Rev., 130, 649–667, doi:10.1175/1520-0493(2002)130<0649:TMCCIS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Neale, R. B., and Coauthors, 2010: Description of the NCAR Community Atmosphere Model (CAM 5.0). NCAR Tech. Note NCAR/TN-486+STR, National Center for Atmospheric Research, Boulder, CO, 289 pp. [Available online at http://www.cesm.ucar.edu/models/cesm1.0/cam/docs/description/cam5_desc.pdf.]

  • 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

  • Ullrich, P. A., P. H. Lauritzen, and C. Jablonowski, 2013: Some considerations for high-order incremental remap-based transport schemes: Edges, reconstructions, and area integration. Int. J. Numer. Methods Fluids, 71, 1131–1151, doi:10.1002/fld.3703.

    • Search Google Scholar
    • Export Citation

  • Zarzycki, C. M., M. N. Levy, C. Jablonowski, J. R. Overfelt, M. A. Taylor, and P. A. Ullrich, 2014: Aquaplanet experiments using CAM’s variable resolution dynamical core. J. Climate, 27, 5481–5503, doi:10.1175/JCLI-D-14-00004.1.

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

  • Boyd, J. P., 2001: Chebyshev and Fourier Spectral Methods. 2nd ed. Courier Dover Publications, 688 pp.

  • Boyd, S., and L. Vandenberghe, 2009: Convex Optimization. Cambridge University Press, 725 pp.

  • Deville, M. O., P. F. Fischer, and E. H. Mund, 2002: High-Order Methods for Incompressible Fluid Flow. Vol. 9. Cambridge University Press, 528 pp.

    • Search Google Scholar
    • Export Citation

  • Dong, L., and B. Wang, 2013: Trajectory-tracking scheme in Lagrangian form for solving linear advection problems: Interface spatial discretization. Mon. Wea. Rev., 141, 324–339, doi:10.1175/MWR-D-12-00058.1.

    • Search Google Scholar
    • Export Citation

  • Dukowicz, J. K., and J. W. Kodis, 1987: Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations. SIAM J. Sci. Stat. Comput., 8, 305–321, doi:10.1137/0908037.

    • Search Google Scholar
    • Export Citation

  • Dunavant, D., 1985: High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int. J. Numer. Methods Eng., 21, 1129–1148, doi:10.1002/nme.1620210612.

    • Search Google Scholar
    • Export Citation

  • Erath, C., P. H. Lauritzen, and H. M. Tufo, 2013: On mass conservation in high-order high-resolution rigorous remapping schemes on the sphere. Mon. Wea. Rev., 141, 2128–2133, doi:10.1175/MWR-D-13-00002.1.

    • Search Google Scholar
    • Export Citation

  • Farrell, P., 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., 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

  • Guba, O., M. A. Taylor, P. A. Ullrich, J. R. Overfelt, and M. N. Levy, 2014: The spectral element method (SEM) 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, 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

  • 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

  • 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

  • Lauritzen, P. H., R. D. Nair, and P. A. Ullrich, 2010: A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid. J. Comput. Phys., 229, 1401–1424, doi:10.1016/j.jcp.2009.10.036.

    • Search Google Scholar
    • Export Citation

  • Margolin, L., and M. Shashkov, 2003: Second-order sign-preserving conservative interpolation (remapping) on general grids. J. Comput. Phys., 184, 266–298, doi:10.1016/S0021-9991(02)00033-5.

    • Search Google Scholar
    • Export Citation

  • Nair, R. D., and B. Machenhauer, 2002: The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere. Mon. Wea. Rev., 130, 649–667, doi:10.1175/1520-0493(2002)130<0649:TMCCIS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Neale, R. B., and Coauthors, 2010: Description of the NCAR Community Atmosphere Model (CAM 5.0). NCAR Tech. Note NCAR/TN-486+STR, National Center for Atmospheric Research, Boulder, CO, 289 pp. [Available online at http://www.cesm.ucar.edu/models/cesm1.0/cam/docs/description/cam5_desc.pdf.]

  • 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

  • Ullrich, P. A., P. H. Lauritzen, and C. Jablonowski, 2013: Some considerations for high-order incremental remap-based transport schemes: Edges, reconstructions, and area integration. Int. J. Numer. Methods Fluids, 71, 1131–1151, doi:10.1002/fld.3703.

    • Search Google Scholar
    • Export Citation

  • Zarzycki, C. M., M. N. Levy, C. Jablonowski, J. R. Overfelt, M. A. Taylor, and P. A. Ullrich, 2014: Aquaplanet experiments using CAM’s variable resolution dynamical core. J. Climate, 27, 5481–5503, doi:10.1175/JCLI-D-14-00004.1.

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

    A simplified depiction of a source mesh, target mesh, and overlap mesh along with associated element indices. Region is local with , , , and .

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

    (a) Geometric distribution of degrees of freedom in a fourth-order Gauss–Lobatto–Legendre finite element. (b) Degrees of freedom in the fourth-order Gauss–Lobatto–Legendre reference element, with coordinate axes and .

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

    (a),(b) Third- and (c),(d) fourth-order GLL basis functions used for the continuous reconstruction: (a),(c) GLL basis and (b),(d) monotone GLL basis.

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

    Subdivision of (a) a quadrilateral and (b) pentagon into triangles.

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

    (a)–(c) First-, fourth-, and eighth-order triangular quadrature nodes.

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

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

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

    (a)–(c) Contour plots of the three test fields used in this study. The fields in (a) and (b) take on values in the range [1, 3]. The field in (c) takes on values in the approximate range [0.46, 1.54].

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

    (left to right) Standard , , 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 latitude–longitude mesh for cubed-sphere resolutions and for three orders of accuracy .

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

    Absolute (a) and (b) 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 latitude–longitude mesh for cubed-sphere resolutions and for three orders of accuracy . Circled data points indicate that the global minimum–maximum has been enhanced (i.e., that monotonicity was not maintained).

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

    As in Fig. 8, but for conservative, consistent, and strictly monotonic remapping.

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

    As in Fig. 9, but for conservative, consistent, and strictly monotonic remapping.

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

    As in Fig. 8, but for remapping to the geodesic mesh for cubed-sphere resolutions and for three orders of accuracy .

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

    As in Fig. 9, but for remapping the geodesic mesh for cubed-sphere resolutions and for three orders of accuracy .

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

    Two “real data” tests for remapping from the variable resolution cubed-sphere mesh (Fig. 6d) with to a 0.25° great-circle latitude–longitude grid (Fig. 6b). (a) Surface pressure from a variable resolution simulation using conservative and consistent remapping. Observe that the detail of the result is much finer between 135° and 90°W in the Northern Hemisphere, corresponding to the region of highest mesh refinement. (b) Percentage plant functional type (barren land) using conservative, consistent, and monotone remapping. This field is highly discontinuous and requires that the data are constrained to the interval to be considered meaningful.

Fig. 1.

A simplified depiction of a source mesh, target mesh, and overlap mesh along with associated element indices. Region is local with , , , and .
Fig. 2.

(a) Geometric distribution of degrees of freedom in a fourth-order Gauss–Lobatto–Legendre finite element. (b) Degrees of freedom in the fourth-order Gauss–Lobatto–Legendre reference element, with coordinate axes and .
Fig. 3.

(a),(b) Third- and (c),(d) fourth-order GLL basis functions used for the continuous reconstruction: (a),(c) GLL basis and (b),(d) monotone GLL basis.
Fig. 4.

Subdivision of (a) a quadrilateral and (b) pentagon into triangles.
Fig. 5.

(a)–(c) First-, fourth-, and eighth-order triangular quadrature nodes.
Fig. 6.

A depiction of the four meshes studied in this manuscript: (a) cubed sphere, (b) great-circle latitude–longitude, (c) icosahedral–geodesic, and (d) refined cubed sphere.
Fig. 7.

(a)–(c) Contour plots of the three test fields used in this study. The fields in (a) and (b) take on values in the range [1, 3]. The field in (c) takes on values in the approximate range [0.46, 1.54].
Fig. 8.

(left to right) Standard , , 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 latitude–longitude mesh for cubed-sphere resolutions and for three orders of accuracy .
Fig. 9.

Absolute (a) and (b) 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 latitude–longitude mesh for cubed-sphere resolutions and for three orders of accuracy . Circled data points indicate that the global minimum–maximum has been enhanced (i.e., that monotonicity was not maintained).
Fig. 10.

As in Fig. 8, but for conservative, consistent, and strictly monotonic remapping.
Fig. 11.

As in Fig. 9, but for conservative, consistent, and strictly monotonic remapping.
Fig. 12.

As in Fig. 8, but for remapping to the geodesic mesh for cubed-sphere resolutions and for three orders of accuracy .
Fig. 13.

As in Fig. 9, but for remapping the geodesic mesh for cubed-sphere resolutions and for three orders of accuracy .
Fig. 14.

Two “real data” tests for remapping from the variable resolution cubed-sphere mesh (Fig. 6d) with to a 0.25° great-circle latitude–longitude grid (Fig. 6b). (a) Surface pressure from a variable resolution simulation using conservative and consistent remapping. Observe that the detail of the result is much finer between 135° and 90°W in the Northern Hemisphere, corresponding to the region of highest mesh refinement. (b) Percentage plant functional type (barren land) using conservative, consistent, and monotone remapping. This field is highly discontinuous and requires that the data are constrained to the interval to be considered meaningful.
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