The RBF method was introduced by Rolland Hardy in the 70s as a new interpolation technique, but it is not until the 90s that Ed Kansa used this method to numerically solve PDEs. It has three major advantages: it is meshfree, easy to implement in any number of dimensions and spectrally accurate for certain types of radial functions. In the context of solving PDEs, its accuracy depends on three things: the type of the radial function, the value of the RBF shape parameter (which controls how steep or how flat a radial function will be), and for time-dependent problems, also the integration time. In this presentation, I will introduce the method in both contexts of interpolation and of solving PDEs. I will then focus on the solution of hyperbolic equations around the surface of the sphere, with a quick overview on the RBF-QR method, introduced by Fornberg and Piret in 2007 to go around an important numerical conditioning issue of the RBF method, encountered in this context.

Earthquakes are manifested always in unexpected manner because of the nature of the nonlinear properties of crustal material, where a critical temperature separates a stable creeping regime from a domain where fracture instabilities can develop. This temperature depends sensitively on the material properties and is different by 200 to 250 degrees, for crustal and mantle substances. Therefore, thermal and chemical effects are important.

We will go over several important concepts in the generation of brittle-ductile instabilities leading to slow earthquakes. We will then review the governing non-linear evolutionary partial differential equations needed to model the brittle-ductile instabilities by taking explicitly thermo-mechanical feedback into account. Then we will also discuss some ongoing laboratory experiments, which will call our attention to the potentially important role played by volatiles and strain-localization in fracturing the crust. Other catastrophic near-surface events , such as landslides and ice-quakes wlll also be discussed within this general framework of instabilites viewed from a physical standpoint.