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Solving PDEs with radial basis functions. (English) Zbl 1316.65073

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.

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
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