3.4.3 Spectral representation and finite element methods

In addition to finite differences, several other methods can be use to discretise equations. One method is to integrate the basic equation of the system over a finite volume before the discretisation. This finite-volume method, has the advantage that it explicitly and easily ensures the conservation of some important properties. If particular hypotheses are made about the changes in the variables inside the volumes, numerical schemes similar to those described using finite-difference methods can be obtained.

In the Galerkin approach, the numerical solution is discretised in space as a sum of k basis functions ${\varphi }_k\left(x\right)$ using coefficients A_k that depend on the time, leading to a spectral representation of the solution:

The goal of the resolution of the problem is then to determine the coefficients A_k. The larger the number of basis functions retained (i.e. the larger k is), the more precise is the spatial representation of the solution. The big advantage of this method is that the space derivation of U(x,t) can be analytically computed from d${\varphi }_k\left(x\right)$/dx without any additional approximation. The choice of the basis function must be adequate. For a one-dimensional problem, Fourier series are a natural choice. For a problem with spherical geometry, in particular for global atmospheric models, spherical harmonics are used. They are the product of Fourier series in longitude and associated Legendre polynomials in latitude.

The spectral and grid-point (also referred to as physical space, see for instance Fig. 3.13) representations are complementary. The first provides an accurate computation of horizontal derivatives while the second forms a convenient framework to evaluate non-linear terms as well as the physics of the model (for example, the computation of the radiative transfer). When the two approaches are combined, some transfers of the variables from the physical to the spectral space are required in the transform method. There must be an exact correspondence between the number of basis function and the horizontal resolution of the grid to avoid numerical problems during this transformation.

The finite element approach is based on a similar approximation to Eq. 3.29 but instead of functions ${\varphi }_k\left(x\right)$ that cover the whole domain as in the spectral method, local basis functions are used. For example, ${\varphi }_k\left(x\right)$ can be a piecewise linear function equal to 1 at a grid point and 0 at all the other points.

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