Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.

The paper derives the first known numerical shallow water model on the sphere using radial basis function (RBF) spatial discretization, a novel numerical methodology that does not require any grid or mesh. In order to perform a study with regard to its spatial and temporal errors, two nonlinear test cases with known analytical solutions are considered. The first is a global steady-state flow with a compactly supported velocity field, while the second is an unsteady flow where features in the flow must be kept intact without dispersion. This behaviour is achieved by introducing forcing terms in the shallow water equations. Error and time stability studies are performed, both as the number of nodes are uniformly increased and the shape parameter of the RBF is varied, especially in the flat basis function limit. Results show that the RBF method is spectral, giving exceptionally high accuracy for low number of basis functions while being able to take unusually large time steps. In order to put it in the context of other commonly used global spectral methods on a sphere, comparisons are given with respect to spherical harmonics, double Fourier series and spectral element methods.

A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.

Axisymmetric force-free magnetic fields external to a unit sphere are studied as solutions to boundary value problems in an unbounded domain posed by the equilibrium equations. It is well known from virial consid- erations that stringent global constraints apply for a force-free field to be confined in equilibrium against expansion into the unbounded space. This property as a basic mechanism for solar coronal mass ejections is explored by examining several sequences of axisymmetric force-free fields of an increasing total azimuthal flux with a power-law distribution over the poloidal field. Particular attention is paid to the formation of an azimuthal rope of twisted magnetic field embedded within the global field, and to the energy storage properties associated with such a structure. These sequences of solutions demonstrate (1) the formation of self-similar regions in the far global field where details of the inner boundary conditions are mathematically irrelevant, and (2) the possibility that there is a maximum to the amount of azimuthal magnetic flux confined by a poloidal field of a fixed flux anchored rigidly to the inner boundary. The nonlinear elliptic boundary value problems we treat are mathe- matically interesting and challenging, requiring a specially designed solver, which is described in the Appendix. Subject headings: MHD — Sun: corona — Sun: coronal mass ejections (CMEs) — Sun: magnetic fields