Fornberg, Bengt; Flyer, Natasha Solving PDEs with radial basis functions. (English) Zbl 1316.65073 Acta Numerica 24, 215-258 (2015). Summary: Finite differences (FD) 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 (RBF), in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale partial differential equation (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. Cited in 117 Documents MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35K57 Reaction-diffusion equations 35Q30 Navier-Stokes equations 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76M20 Finite difference methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis Keywords:Navier-Stokes equation; survey paper; numerical example; Poisson equation; reaction-diffusion equation; flow in spherical shell; shallow water equation; finite difference; spectral method; radial basis function PDFBibTeX XMLCite \textit{B. Fornberg} and \textit{N. Flyer}, Acta Numerica 24, 215--258 (2015; Zbl 1316.65073) Full Text: DOI