Multidimensional Cases

3 Multidimensional Cases

In principle, the generalization to more than one dimension is rather straightforward if one uses the tensor product. Let us first take an example, with the spectral representation of a scalar function $f(x, y)$ defined on the square $(x,y ) ∈ [− 1,1] × [− 1,1]$ in terms of Chebyshev polynomials. One simply writes

with $Ti$ being the Chebyshev polynomial of degree $i$ . The partial differential operators can also be generalized as being linear operators acting on the space $ℙM ⊗ ℙN$ . Simple linear partial differential equations (PDE) can be solved by one of the methods presented in Section 2.5 (Galerkin, tau or collocation), on this $M N$ -dimensional space. The development (88

) can of course be generalized to any dimension. Some special PDE and spectral basis examples, where the differential equation decouples for some of the coordinates, will be given in Section 3.2.

3.1 Spatial coordinate systems
  3.1.1 Mappings
  3.1.2 Spatial compactification
  3.1.3 Patching in more than one dimension
3.2 Spherical coordinates and harmonics
  3.2.1 Coordinate singularities
  3.2.2 Spherical harmonics
  3.2.3 Tensor components
3.3 Going further
  3.3.1 More than three spatial dimensions