Conclusion

In this chapter, we first described the time integration scheme which is used in the COSMO weather prediction model. We then studied the spatial discretisation used to calculate the tendencies due to diffusion. We compared this method to other similar finite difference methods. We conducted this analysis on two test domains and two different types of grids were used. The first type of grid is equidistant and reproduces only one of the main characteristics of the COSMO grid namely that the lower level is not horizontal. The second type of grid resembles the COSMO grid more since it has the additional property that the distance between the horizontal levels increases quadratically. On domains with a constant slope and when the grid is equidistant, we proved that the scheme is of order one. However, when solving some test cases using smooth functions, we observe order two. The second grid type also leads to second order convergence when smooth functions are used and when Dirichlet boundary conditions are imposed on a domain with constant slope. However, due to the fact that the metric terms are not differentiable at the lowest level, one has to be careful as to the choice of the domain considered when using mixed boundary conditions. When domains defined such that the lowest level is not differentiable everywhere, we showed that second order convergence can be recovered if the exact solution of the problem considered is zero at the points where the function is not differentiable. However, this is seldom the case in NWP so reduction of the order of convergence is to be expected, with error peaks at the corners of the domain.

Chapter 4

The Discrete Duality Finite Volume Method

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

John Von Neumann,First national meeting of the Association for Computing Machinery, 1947 We have seen in Chapter 1 that the Finite Volume Method is more appropriate than the Finite Difference Method for complex geometries. However, the FVM is limited to admissible meshes i.e.

meshes such that the vector connecting two nodes is perpendicular to the edge between those nodes.

A lot of work has been done in the past decades to approximate the gradient in such a way that the orthogonality condition is not necessary. In this chapter, we focus on theDiscrete Duality Finite Volume (DDFV) method. The DDFV method found its origins in the work of Hermeline in 1998 [Her98]. Domelevo and Omnes immediately started working on this new method and presenting it at conferences. Andreianov, Boyer and Hubert took interest and also started working on this method [ABH04]. With more and more people working on it, the new method needed a name so it was baptised in 2005 in [DO05]. The work on the DDFV method has been in full expansion ever since (see for instance [DDO07], [BHK08], [Del09], [AHK11], [ABH13], [CHKM14], [DO15], [Gan+16], [GKL17]).

The numerical scheme considers values at supplementary unknowns, which implies that an equival-ent number of supplemequival-entary equations have to be taken into account. The final system of equations therefore is bigger than the one obtained when only the original values are considered and is slightly more expensive to solve in terms of computational costs. However, the many advantages of DDFV easily compensate for this drawback. The main advantage of this method is that it can be defined on almost arbitrary grids. It is by definition symmetric and leads to a natural definition of discrete operators such as the divergence and the curl.

In the context of numerical weather prediction, it is desirable to consider discrete gradient operators which are curl-free i.e. such that the curl of the gradient vanishes. This is explained by Thuburn and Cotter in [TC12] as follows: “The pressure gradient is one of the largest terms in the governing equations, so small relative errors can have a comparatively large effect. It is particularly important that the pressure gradient should not act as a spurious source of vorticity, since that would contaminate the meteorologically important vortical signal. Thus, a discrete analogue of∇ × ∇η≡0 for any scalar fieldη is needed.”

4.1 Notations

We now give the notations which we use to define the DDFV method and which are exemplified in Figure 4.1.1. Theprimal meshforms a partition of Ω and is composed of I elementsTi. With each elementTi we associate aprimal nodeGi located insideTi. The functionθ_i^T is thecharacteristic functionof the cell Ti. We denote by J the total number of sides of the primal mesh, and by J^Γ

Sk9

S_k8

Sk7

Sk6

S_k5

Sk13

Sk12

Sk11

Sk10

Sk4

Sk2

Sk1

Sk3

Gi1

Gi2

Gi3

Gi4

G_i5

Gi6

Gi7

Gi8

Gi9

Gi18

Gi10

G_i11

Gi12

Gi13

Gi14

Gi15

Gi16

Gi17

Pk2

Ti5

Dj Aj

Figure 4.1.1: Notations for the DDFV method.

the number of these sides which are located on the boundary. We denote thesides of the primal meshbyAj, and assume that they are ordered so thatAj ⊂Γ⇔j∈ {J−J^Γ+ 1, J}. We introduce additional primal nodes to each boundaryA_j, denoted byG_i withi∈ {I+ 1, ..., I+J^Γ}. The nodes of the primal mesh, thedual nodesare denoted by S_k withk∈ {1, .., K}. To eachS_k, we associate adual cellP_k obtained by joining the points G_iassociated with the elements of the primal mesh of whichS_k is a node. The dual mesh also forms a partition of Ω and its sides are denoted byA⁰_j. We assume thatS_k ∈Γ if and only ifk∈ {K−J^Γ+ 1, ..., K}.

To each A_j we associate a diamond-cell obtained by joining the nodes of A_j with the primal nodes associated with the primal cells which share the side Aj (see Figure 4.1.2). The unit vector normal to Aj is denoted by nj and is oriented such that hG_i2(j)−G_i1(j),nji ≥ 0. Similarly, the unit vector normal to A⁰_j is denoted by n⁰_j and is oriented so that hS_k2(j)−S_k1(j),n⁰_ji ≥ 0. For all i∈ {1, . . . , I},j ∈ V(i) (resp. k∈ {1, . . . , K},j ∈ E(k)) we definesji (resp. s⁰_jk) to be 1 ifnj points outward ofTi and -1 otherwise (resp. 1 ifn⁰_j points outward of Pk and -1 otherwise). We thus can define theoutward pointing unit normal vectorsnji=sjinj andn⁰_jk=s⁰_jkn⁰_j.

We defineV(i) :={j∈ {1, . . . , J} |Aj⊂Ti}andE(k) :={j∈ {1, . . . , J} |Sk∈Aj}.

Each diamond cell can be split into two triangles in two distinct ways if the diamond cell is convex and in one way if it is non-convex (see Figure 4.1.2).

Definition 4.1.1 If D_j is convex, we define Tj,γ to beD_j,γ or D_j,γ⁰ andτ_j to ben_j or n⁰_j according to the chosen split. IfD_j is non-convex, then Tj,γ=D_j,γ andτ_j =n_j.

Dj

Figure 4.1.2: Splitting possibilities for diamond cells.