Solving Hyperbolic Partial Differential Equations in Spherical Geometry with Radial Basis Functions

Mathematical modeling of space and climate phenomena generally requires the solution of partial differential equations (PDEs) inside/outside or on a sphere. A key difficulty is that it is impossible to uniformly distribute more than 20 points on a sphere, in contrast to trivially placing any number of points uniformly along the periphery of a circle. The essence of the problem is one that has faced cartographers for at least a millennium the surface of a sphere cannot be mapped to a rectangle without severe distortions and a singularity in at least one location. Similarly, today’s computational mathematicians are constantly grappling with the issue of how to devise efficient numerical methodologies for spherical domains, especially given the advent in the last decade of multi-processor distributed memory computers that are needed for large-scale simulations. Currently used numerical methodologies all have specialized strengths but also serious weaknesses, mostly due to their association with an underlying grid. Global methods, such as double Fourier series or spherical harmonics, do not practically allow for local mesh refinement. The latter also involves cumbersome algebra, and can not be efficiently parallelized on multiprocessor machines. Localized methods, such as spectral element methods, have to refine near unphysical interior boundaries (due to element discretization as well as mapping onto a cube) in order to suppress Runge phenomena (violent oscillations near edges typical of high-order polynomial interpolation) and involve high algorithmic complexity. As a result, computational scientists in the field of climate and space modeling are scrambling for new options. Radial basis functions(RBFs) offers a completely new numerical approach to the scientific community for solving time-dependent PDEs in spherical domains. This new methodology has the beauty of being grid-independent, allowing for non-uniform resolution, while providing spectrally accurate solutions to PDEs in multi-dimensions. However, the field is still in its development stage. The application of RBFs to pure hyperbolic systems has never been successfully addressed while the simpler problems of solving linear elliptic or parabolic PDEs in spherical geometries having only been addressed in the last three years. The few articles in the field are mainly of a theoretical nature, containing little or no computations or applications to physical modeling. In the current presentation, we will give a brief overview of commonly used high-order methods for modeling spherical geometries, placing RBFs in this context. Next, we will derive the RBF formulation of the gradient operator in spherical coordinates that will be needed for solving the linear advection equation on a sphere. A method of lines formulation will be developed, illustrating how the discrete spatial differential operator is free of any singularities at the pole that are present in the original PDE when posed in spherical coordinates (and also present in other pseudospectral methods such as spherical harmonics). Secondly, the spectral accuracy of infinitely smooth RBFs will be demonstrated, showing high spatial accuracy for much lower resolution as compared to other spectral methods. In the oral presentation, time stability will be further analyzed along with other test cases.