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Radial basis functions: A mesh-free modeling framework for computational geosciences

Mathematical models for geophysical phenomena are typically formulated as some system of partial differential equations (e.g., Stokes’ equations, shallow water wave equations, Richards equation, elastic wave equations, and Maxwell’s equations).  Solving these equations can rarely be achieved through analytical means (i.e., pen and paper) and one must instead employ numerical techniques (i.e., high performance computers).  This requires choosing a framework for discretizing the continuous system of equations that is amenable to certain features of the model, such as equation type (hyperbolic, parabolic, or elliptic), spatial dimension, differential/integral relationships, non-linear terms, smoothness of solutions, and domain geometry.

In this talk I will discuss a relatively new numerical modeling framework based on radial basis functions that has many attractive features over conventional methods currently in use for computational geosciences.  For example, radial basis functions are mesh-free, meaning they require no computational grid or mesh like finite difference, finite volume, finite element, discontinuous Galerkin, spectral element, and spectral/pseudospectral methods.  This makes radial basis functions particular useful for applications that feature irregular geometries, or applications where measured or observed data is not given on a nice grid/mesh.  Additionally, they can be customized to preserve certain differential/integral constraints present in the physical models.  I will focus on these aspects of radial basis functions and present some applications for global atmospheric flows, 3D mantle convection in the earth’s interior, and laboratory experiments on baroclinic instability mechanisms.  I will conclude with possible future applications to seismic and radar imaging.