10.25560/93774
Snowball, Benjamin Samuel Thomas
Imperial College London
Sparse spectral methods on disk-slices, trapeziums and spherical caps
Imperial College London
2021
Olver, Sheehan
Cotter, Colin
Engineering and Physical Sciences Research Council
2021-12
2021-06
2022-01-20
Doctor of Philosophy (PhD)
10044/1/93774
Creative Commons Attribution-Non Commercial 4.0 International Licence
https://creativecommons.org/licenses/by-nc/4.0/
This thesis develops sparse spectral methods for solving partial differential equations (PDEs) on various multidimensional domains, with a specific focus on the disk-slice and trapezium in 2D, and the spherical cap in 3D. For the latter, the PDEs are surface PDEs involving Laplace-Beltrami operators, spherical gradients and other spherical operators. We begin with an introduction to sparse spectral methods via viewing spherical harmonics as multidimensional orthogonal polynomials in x, y, and z. We explain how differential operators can be applied as banded-block-banded matrix operators to coefficient vectors for a function’s expansion. Further, we demonstrate how vector spherical harmonics in x, y, and z can be used as an orthogonal basis for vector-valued functions, yielding similar banded-block-banded gradient and divergence operators. We move on to presenting a new framework for choosing a suitable orthogonal polynomial basis for more general 2D domains defined via an algebraic curve as a boundary. This work builds on the observation that sparsity is guaranteed due to this definition of the boundary, and that the entries of partial differential operators can be determined using formulae in terms of (non-classical) univariate orthogonal polynomials. Triangles and the full disk are then special cases of our framework, which we formalise for the disk-slice and trapezium. Piecing together the techniques used thus far, we present a new orthogonal polynomial basis and sparse spectral method for the spherical cap, complete with the same observation of the guaranteed sparsity of operators. The motivation is for one to use spherical caps bands as in a spectral element method for the sphere, with many applications in meteorology and astrophysics – in particular, as a potential replacement of the spherical harmonics approach currently in use at ECMWF, which is predicted to become too costly due to a parallel scalability bottleneck arising from the global spectral transform.