# Short Review of Current Numerical Developments in Meteorological Modelling

## Abstract

**:**

## 1. Introduction

## 2. Differentiation Stencils of Three, Four and Six Legs and Diagnostic Legs

## 3. The Interface for the Connection of L-Galerkin Methods to a Background Model

## 4. Cut Cells as an Example for the Use of Geometric Files

## 5. Ten Scientific Project Steps towards a Realistic Forecast Model Based on Sparse L-Galerkin Method of Basis Function Orders Two, Three, and Four

- (1)
- Establish grid properties of polygonal models on the sphere in a systematic way: Polyhedral grids are based on the cube (cubed sphere), the icosahedron and the T36 solid (examples of this paper). The creation of such polyhedral models normally starts with the representation of a grid similar to that in Figure 2. When the creator is optically satisfied with its homogeneity, the program is created and the modeler hopes for sufficient stability and a reasonable CFL number. To the author’s knowledge, there exists no systematic evaluation of grid properties for the different grids based on the different polyhedral solids. It is highly desirable to create such systematic evaluations dependent on the resolution. Grid properties of interest are grid length, angles of stencils and homogeneity. The latter means the difference of good lengths in two different directions. This information can be used to get a priori information on the CFL, and badly designed stencils may be enhanced by a diagnostic stencil leg in order to enhance the CFL.
- (2)
- Solve the spectral gap/0-space problem for basis function representations of third order: The spectral element scheme has a small amplitude feature with negative group velocities at the 6 dx wave. The o3o3 has a large 0-space up to 4 dx. It seems likely that these two features are caused by the computation of the derivative at corner points. In fact, for o3o3, first experiments indicated that there is a large sensitivity of this phenomenon to different ways of computing the derivative at corner points. For o3o3, this means that the effective resolution is only that of the o2o3 scheme, still achieving a degree of sparseness and the corresponding saving of CPU time. For optimum efficiency of the two L-Galerkin schemes, it is desirable to solve these problems. There is no shortage of variants of these schemes which are worth trying. Without this achievement, it is still possible to go ahead with L-Galerkin schemes. The spectral element schemes are already in practical use (see St_Li for a review). For o3o3, it seems best to change to o2o3 currently and accept the smaller degree of sparseness. The totally irregular grid can be approached with o2o2. If the grid is formed by great circles, such as icosahedral grids, o2o3 or o3o3 can be applied. The spectral gap problem needs to be solved when a fourth- or third-order treatment is desired.
- (3)
- Approach the totally irregular grid for second-order methods. For test purposes, totally irregular grids can be approached as structured grids (see St_Li). Otherwise, a memory management scheme is necessary, as used, for example, with MPAS [5] or Fluidity [11,13]. In second-order approximation, this can be done using the tools provided in St_Li. While there is no doubt about the possibility of such applications, currently there exist no tests and these are necessary before going to applications such as Fluidity. As shown in Appendix B, the current schemes of Fluidity are somewhat below order two, and therefore second-order developments are still up to date. Due to the approximation order paradox (see St_Li), there is currently no benefit for forecast quality when going to third order with realistic models, even though toy models show such benefits very strongly. Even high-order models such as MPAS [5] are no exception due to problems of vertical discretization. Therefore, it is still interesting to improve second-order methods in practical modelling.
- (4)
- Test the 3:1 or 2:1 interface for physics interfaces. In St_Li, it was argued that with third-order methods, it would be appropriate to call the physical parameterizations only every third point. For second-order schemes, such as o2o2, and for o2o3, this would mean calling them every second grid point. Very simple 1D tests were given and it would be highly desirable to test this with realistic 3D models.
- (5)
- Test rhomboidal methods on the sphere. The Williamsson shallow water tests [13] should be conducted to gain confidence in the new L-Galerkin methods. With rhomboidal discretization, there is no problem to do this also for o3o3 or o2o3, keeping in mind the limitations of o3o3 due to the 0-space problem.
- (6)
- Test hexagonal methods on the sphere. Hexagons on the plane were tested. The program used there was so complicated that it was decided that further progress is not possible without a better compact storage for the amplitudes on the hexagon. The storage associated with the centers of the hexagonal cells is assumed to be suitable. Note that the center point stores the amplitudes but does not have an amplitude itself. The rather limited hexagonal tests given before are to be extended to the Williamsson tests.
- (7)
- Conduct 3D tests with the cut-cell scheme in second order. The quick and dirty scheme used earlier should be replaced by a conserving scheme such as that described in Section 4.
- (8)
- Extend Point (7) to third order and extend Point (5) to three dimensions on the sphere. This can be done using a resolution on the sphere of 200 km and it is expected that this would fit into a PC. For comparison, the non-conserving classic o4 method with terrain-following coordinates should be implemented and compared to the rhomboidal method for efficiency.
- (9)
- With a resolution of 5 km on the sphere, carry out one realistic case and evaluate meteorologically.
- (10)
- For the model of Point (9), carry out a series of 100 cases and evaluate statistically.

## 6. Simple Linear Tests to Be Used with Irregular Resolutions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Programs for Atmospheric Modelling on PCs

- -
- A Fortran compiler;
- -
- A plot routine drawing the line between two given points;
- -
- A plot routine to draw curves;
- -
- A plot routine for isolines.

## Appendix B. The Order of Approximation of Models Based on the Finite Volume Method

## Appendix C. The Loss of Super-Convergence of the Classic Galerkin Finite Element Method with Irregular Resolution

- (1)
- The finite volume method is defined as:

- (2)
- The finite element classic Galerkin approximation of the derivative $\frac{\partial {f}_{j}}{\partial x}$ of a function f(x) uses functional approximations of f and $\frac{\partial {f}_{j}}{\partial x}$ where $f\left(x\right)={\displaystyle {\sum}_{j}{f}_{j}{e}_{j}\left(x\right)}$ and $\frac{\partial f}{\partial x}={\displaystyle {\sum}_{j}\frac{\partial {f}_{j}}{\partial x}{e}_{j}\left(x\right)}$. In the above expressions, ${e}_{j}\left(x\right)$ is the piecewise linear hat function, being 1 at ${x}_{j}$ and 0 at all other grid points. The Galerkin approximation equation is:$$\sum}_{j}\frac{\partial {f}_{j}}{\partial x}{e}_{j}\left(x\right)}={\displaystyle {\sum}_{j}{f}_{j}\frac{\partial {e}_{j}\left(x\right)}{\partial x$$

**Figure A1.**Error of the computation of the derivative function of dx for the classic Galerkin method for regular and irregular grids. With regular resolution, we have super-convergence to fourth order and for irregular resolution, we have first-order convergence.

## Appendix D. Convergence of Advection with an Irregular Mesh in the Model Fluidity

2 m | 4 m | 8 m |
---|---|---|

0.049 | 0.072 | 0.111 |

**Figure A2.**The advection tests along a straight-line mountain using DG; the error at time 1 s and 100 s is shown. This error is concentrated near the position of the cloud, which for some of the less performing schemes of Fluidity is not the case. The interest of this test is the decrease in the error with increasing resolutions. This is shown in Table A1 and it shows linear convergence, where with regular grids (away from the surface) we see second-order convergence.

**Figure A3.**A supercell simulation by the model Fluidity, showing relative moisture (red = 100%, white = 0%). A cross section at 500 m is shown.

**Figure A4.**Advection of a 2D cloud along a curved lower boundary. Note that we have the slip boundary, where the velocity is different from 0 at the boundary. The orography has the form of a ramp and the velocity at the surface is parallel to this ramp. The advection of the cloud follows this ramp. This indicates an error much smaller than that shown in Figure A2, even though the situation is more difficult, not being a straight line.

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**Figure 1.**Second-order differencing in the grid ${x}_{i}$. The polynomials used for the differentiation of second approximation order at different grid points A, B, C. For second-order differencing at points A and C, the second-order polynomials are indicated as solid lines. The second-order differentiation at point B is carried out using the polynomial indicated as a dotted line. In contrast to this, a basis function method of second-order representation for h uses a unique interpolation, indicated as a solid line. It appears that direct differentiation is not possible at point B. The methods o2, spectral elements and the L-Galerkin method o2o2 use other means than direct differentiation to obtain time derivatives at such points. With spectral elements, points A and B are called inner collocation points and the time derivatives at such points are computed in the spectral representation (see St_Li). The objective of high-order differencing is to obtain an accuracy higher than second order, popular choices are third or fourth order. The construction of polynomials in higher than second order is similar as with second order, but more complicated. Basis function methods, such as spectral elements, o3o3 or o2o3, combine this with conservation properties, which are not present with classic o4. In addition, o3o3 and o2o3 support sparse grids, with the corresponding improvement of CPU times.

**Figure 2.**Right (

**b**): A rhomboidal cell grid in the T36/R18 solid. The T36 at the basic of this construction carries 18 rhomboidal basic spherical rhomboids, of which 6 are around the north pole and 6 around the south pole. The spherical rhomboids around the north and south poles are identical and regular. The six spherical rhomboids at the equator are also identical and regular, but not identical to the rhomboids around the poles. Therefore, the T36 is not a Platonic solid, but rather semi-Platonic. The cell structures derived from Platonic solids will be semi-Platonic, even when cubes or the icosahedron is taken as a basis. Therefore, it may be permitted to choose the basic rhomboidal structures to be semi-Platonic. Each spherical edge of a basic rhomboid is divided into nine grid intervals to obtain the cells. The left panel (

**a**) shows a hexagonal cell structure. Triangular cells are obtained by dividing each rhomboidal cell into two triangles. For the sparse grids, the points are the corners and edges of the cells. The triangular grid is the full grid, containing all corner points of a cell structure, and this resolution is three times increased when introducing the points. Sparse grids are obtained by treating some of the points diagnostically. The right panel (

**b**) shows a cell structure with a cell length of 1/3 of the edge of a basis rhomboid. The hexagonal grid has a reduced number of grid points, making the grid sparse. Grid points are on the edges or corners of the hexagons. The hexagonal centers do not carry amplitudes. The amplitude-carrying points in the right panel (

**b**) are the crosses on the corners and edges of the hexagons. Only the points for one large rhomboid are shown. If the grid points for the other basic rhomboids are drawn, a full hexagonal sparse grid is created. This means that the points shown in the left panel (

**a**) taken for all basic rhomboids of the basic solid form a compact set. The index of a point has four values, ${i}_{rhc}$ i, j, ${i}_{poi}$. ${i}_{rh}$ is the index of the panel, i, j is the index of the hexagonal center and ${i}_{rh}$ = 1, 2, 3, 4, 5, 6, 7, 8 indicates the position of the amplitude within each hexagon. h therefore has four indices: h${i}_{rh}$, i, j, ${i}_{poi}$. The range of these indices is ${i}_{rh}$ = (1, 2, …, 18), I = 1, …, ie, j = 1, …, ie). ie determines the resolution of the grid. In the left panel (

**a**), we have ${i}_{edge}$ = 1. Compact storage means that within the range of ${i}_{edge}$, i, j and ${i}_{edge}$, all values have amplitudes. Compact storage is convenient for all computers and necessary when a great deal of message passing is required.

**Figure 3.**A cell for 3D discretization. Each volume contains more than one grid point. The grid points form the collocation grid. A sparse grid is shown. The points for amplitude storage are indicated with black color. The white points also belong to this cell but are stored with other cells. There are 4 × 4 × 4 points in a cubic cell, some of them also belonging to other cubic cells. Of these, there are 3 × 3 × 3 = 27 independent points for amplitudes. The grid points shown for the sparse grids are only seven. Each cubic cell is indexed by i, j, k, and the points inside the cube are indexed by ${i}_{p}$ = 0, 1, 2, 3, 4, 5, 6. The total index is h

_{i,j,k}, ${i}_{p}$. This representation provides a compact storage. This means that h is dimensioned by (0:ie, 0:je, 0:7). This is a compact storage, as all indices are used and carry amplitudes. An alternative is to dimension h by (0:3 × ie, 0:3 × je, 0:3 × ze). The figure above shows that for the second representation many points are unused. The second representation is therefore non-compact, and for computer representation, this is a severe disadvantage, as message passing may occur for points not carrying an amplitude.

**Figure 4.**Four- and three-legged stencils with corner points (black dots and edge points (+)). Some of the legs are extended by dotted lines to straight lines. (

**a**) A 4-legged stencil where two pairs of legs extend to form two straight line stencils. For this situation, Equation (3) is directly applicable to form derivatives in the x and y directions. (

**b**) Same as (

**a**), but no leg aligns with another. (

**c**) The angles $\alpha $, $\beta $ and $\gamma $ of a three-legged stencil. These angles will determine the accuracy and stability of schemes based on such a three-legged stencil. The three angles may range from slightly above 0 to $\pi $. Obviously, when one of the angles is near 0 or $\pi $, the result for the derivative is very inaccurate. Until now, there exists no investigation of the accuracy and stability of derivatives taken from different stencils. Also, the distributions of such angles in hexagonal grids, as shown in Figure 2, has not yet been investigated. (

**d**) shows as dotted lines additional diagnostic stencil lines which can be used to make the stencil a straight-line stencil and to improve the regularity of stencils such as shown in Figure 4. A three-legged stencil of the kind where extremely low accuracy is expected is shown in (

**e**).

**Figure 5.**Piecewise linear (

**a**) and differentiable (

**b**) lower boundaries and their effect on the cell boundaries. For the left panel, if u is a continuous function, then v must necessarily be discontinuous when requiring the velocity vector on the mountain to be in the direction of the mountain. Discontinuous basis functions are used in connection with the Arakawa C-grid. In the left panel, the white points are amplitudes for the definition of the density field h and + indicates definition points for the velocity points $u$ or $w$. $u$ is piecewise constant in the y direction and linear in the $x$ direction, and vice versa for $v$. $h$ is piecewise constant on the cell areas. Black points are node points of the orographic spline. The right panel shows a piecewise differentiable orographic spline. This is the discretization to be preferred with high-order L-Galerkin schemes, either spectral elements or o2o3. Dashed lines show the division of cut cells into triangles for the case. In the Fluidity model, corner amplitudes are indicated as white or green points (when they are on the orographic line). For the sake of numerical efficiency, some of the amplitudes may be treated diagnostically.

**Figure 6.**Advection of the point initial condition by the o3o3 method. Left: plot of the point initial values. Right: plot after an advection over 400 grid intervals. For this very small-scale initial value, the 0-space shows as a structure remaining at the position of the initial condition and not being transported. This shows the 0-space. The non-transported feature is formed by the 0-space. Such non-transported parts of the solution vanish when the initial condition is smooth and involves more than one grid point, meaning initial values derived from Equation (7). The transported part of the solution, though showing strong dispersion and a loss of amplitude, has some forecast pow-er. (

**a**) shows the initial values and (

**b**) the forecasted field.7. Conclusions.

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**MDPI and ACS Style**

Steppeler, J.
Short Review of Current Numerical Developments in Meteorological Modelling. *Atmosphere* **2024**, *15*, 830.
https://doi.org/10.3390/atmos15070830

**AMA Style**

Steppeler J.
Short Review of Current Numerical Developments in Meteorological Modelling. *Atmosphere*. 2024; 15(7):830.
https://doi.org/10.3390/atmos15070830

**Chicago/Turabian Style**

Steppeler, Jürgen.
2024. "Short Review of Current Numerical Developments in Meteorological Modelling" *Atmosphere* 15, no. 7: 830.
https://doi.org/10.3390/atmos15070830